TL,DR: cold fusion is real, apparently. Yes, really - cold fusion. I know. I wouldn't have thought so either.
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The point of this post is basically to promote to your attention the hypothesis that cold fusion is a real physical phenomenon. For those of you not in the know, this very much flies in the face of current scientific consensus (something I'm not usually in the habit of opposing). In this case though the evidence seems to be quite straightforwardly in favour of the cold fusion advocates.
[Note: most researchers working in this area don't like the term cold fusion; partially because of the negative scientific connotations it drudges up, and partially because fusion might not be an accurate description of what's going on physically. The two preferred terms seem to be low-energy nuclear reactions (LENR) and lattice-assisted nuclear reactions (LANR). I use cold fusion in this piece mainly for convenience and name-brand recognition]
Quick background - in 1989 Stanley Pons and Martin Fleischmann, leading electrochemists of their day, announced a truly startling discovery: a tabletop apparatus of theirs had produced anomalous heat that was (according to them) orders of magnitude beyond what could be produced by chemical effects alone. The only process that can produce heat like that is a nuclear reaction, but such reactions were thought to be impossible at such low temperatures. Thinking they had discovered a new source of energy, Pons and Fleischmann were justifiably excited and hurried to publish their results. In the subsequent months a huge number of researchers tried to replicate their findings, with most being unable to do so. Of the few scientists who did get positive results, some later retracted their work, and others were criticized for sloppy experimental design. To make matters worse, errors and exaggerations were found in Pons and Fleischmann's original paper. Very quickly the scientific community as a whole had cold fusion pegged as "pathological science", and most researchers forgot about the whole affair and went back to their normal, non-energy-crisis-solving work. Pons and Fleischmann, disgraced, ended up quietly leaving the country to continue their work elsewhere, and that was the end of the cold fusion story, as far as most people were concerned. 
Here's where it gets interesting. Naturally, the prospect of solving the world's energy problems proved very alluring to people, so a small number of researchers continued their work with cold fusion. During the 90's some of this work was published in peer-reviewed journals, although this became less and less common as the decade wore on. As far as I know, no mainstream peer-reviewed scientific journal currently accepts cold fusion papers for consideration. Undeterred, cold fusion researchers continued their work; research was published at conferences devoted to cold fusion, self-organized by researchers in the field. This work was generally not peer-reviewed, and much of it (I think most cold fusion researchers would be willing to admit) was not of the highest quality, scientifically. Much - but not all, mind you. There were some researchers at respected universities (including MIT) that conducted very rigorous and high quality studies. Anyway, together this motley band of hobbyists, engineers and scientists, over the last twenty years or so, has found...well, something. Sometimes. If you squint right.
Basically there are a huge number of scattered reports of cold fusion occurring, but reproducibility is a big problem. Some people find low levels of excess heat. Some people find nothing. Some people, when conditions are "just right", report extremely high levels of excess heat. There are even a few cases where explosions occurred and labs have been "blown up" . The sheer volume of claims might be enough to be suggestive that something was going on, all things being equal. But of course, all things aren't really equal in this case; given the initial inability of expert scientists to replicate the original findings in 1989, and the non-peer-reviewed nature of most cold fusion work nowadays, we have every reason to be extra skeptical of reports of cold fusion. Extraordinary claims and all that.
This is why I've taken what I consider to be the two strongest pieces of evidence for cold fusion and provided them below. As I mentioned before, there are some scientists doing rigorous, very well controlled experiments at research universities, and they consistently find that cold fusion is occurring. So, without further ado, here's my proof:
1. Mitchell Swartz's experiments
If you have the time, I would strongly suggest you watch this video: http://www.youtube.com/watch?v=e38Y7HxD_5Y. It's part of a lecture from a multi-week cold fusion course put on by Swartz and others at MIT in January. It's 40 minutes long (and only the first part of five videos actually) but well worth your time. In it Swartz basically makes his case for cold fusion.
I suppose I should stop here to briefly describe what a typical cold fusion experiment looks like. The standard design uses a simple piece of metal, usually Palladium or Nickel. Heavy water (Deuterium) is forced into the lattice of atoms that make up the metal by applying an electric field. Once a high enough loading of Deuterium is achieved in the metal lattice, what (purportedly) happens is that two Deuterons combine in a nuclear reaction to produce a single Helium(4) nucleus, plus heat. The idea is that the lattice of metal atoms is mediating the nuclear reaction in some way, making it occur at far lower temperatures than would normally be possible. Typically these experiments are done with the apparatus fully immersed in heavy water, and what you do is check for excess heat by setting up a calorimeter around the experiment. You can easily measure how much electrical energy you're putting into the system; if the calorimeter is reading higher energies than that coming out, you know cold fusion is taking place (well, that's not entirely accurate - you know some process is producing extra energy, but you don't know what it is. The reason we can confidently say it's nuclear in origin is because the energy densities involved are well beyond what could be produced by a simple chemical reaction).
Anyway, if you can't watch the video, here's what Swartz has found:
-Consistently measures output energy in the range of 200-400% of input energy (!)
-Excess heat is well above noise level for calorimeter
-Calorimeter is very well calibrated - when heat is fed into system via simple ohmic resistor, measured output heat exactly matches input energy
-Chemical control experiments fail (ie using non-cold-fusion-active metals and loading materials gives no excess heat)
-Two calorimeters (each of which have several redundant ways of measuring heat anyway) were built, just to be sure; same results
-Excess heat generation occurs for days or even weeks continuously
-He(4) production is observed, with amounts commensurate with heat production
Mind you, this is not just a one-off experiment - he's been getting results like this for ten years or more. If you watch the video, I think you'll agree that it's a very well-controlled and well-calibrated experiment. It certainly looks that way, anyway, to my semi-informed eyes as a physics grad student (although if there are any actual experimentalists reading this who are more informed than I, I would love to hear from you - please, attack it to bits). In my eyes the only two reasonable explanations for Swartz's results are (i) cold fusion being real, and (ii) active fraud. Fraud is of course possible, but I think unlikely given what other groups have found.
Oh, and if you can't watch the video, here's a 2009 paper you can read by Swartz: http://world.std.com/~mica/Swartz-SurveyJSE2009.pdf. It's less focused on his own research and more of a survey of cold fusion research in general, but he does talk about his own results in Section 4. Certainly worth a look.
2. Yasuhiro Iwamura's transmutation work at Mitsubishi
In one of those strange quirks of fate, for some reason or another scientists in Japan ended up being particularly open to cold fusion claims . There are currently several researchers in Japan, some at universities and some at different companies, who are looking in to cold fusion. I link you here to a particularly interesting paper by Iwamura, who works for a research division of Mitsubishi: http://newenergytimes.com/v2/conferences/2012/ANS2012W/2012Iwamura-ANS-LENR-Paper.pdf
Iwamura uses a slightly different setup for Swartz, but the basic idea is the same: Deuterium is permeated through a Palladium lattice, magic happens, heat comes out, etc. The main difference in this experiment is that Iwamura is not actually looking for excess heat production. He's instead looking for transmutation of elements, which also has been reported to happen in certain cold fusion experiments. To do this a layer of some other material, in this case Cesium, is added on top of the Palladium, and - in a process that no one fully understands yet - that material is transmuted into an entirely different element. So just in case unlimited clean energy wasn't enough for you, we now also have just straight-up alchemy happening (I for one can't fathom why scientists are skeptical of cold fusion).
But, prior probabilities be damned, Iwamura has actually gone and done this! In his experiments he does time-resolved XPS spectroscopy, and observes Praseodymium being created in the apparatus while the total amount of Cesium goes down with time - elemental transmutation (!)
This work is particularly strong evidence for two reasons, I think:
One, because the claim involves detecting elements, it's inherently more plausible than any claims to do with excess heat. Calorimetry can be difficult, and it's easy for a skeptic to claim that the experimenter simply made a mistake in measuring the excess heat (mind you in the case above I think the calorimetry is well done and that there wasn't a mistake, but that isn't always the case). In contrast to calorimetry, detecting elements is very straightforward. There are many independent ways to do it, and it's all rather black and white; either you find an element, or you don't. If you do find a new element, then have something of a smoking gun - it's very difficult to explain how a new element could just appear in your experiment without invoking nuclear processes. The standard skeptic's reply to experiments like this is basically to say "contamination," and wave their hands. That is, they posit that the transmuted element in question was already present in the Palladium lattice at the start of the experiment (perhaps concentrate somewhere so it wasn't detected initially). I find this a less than compelling argument, to say the least - really, the experiment just happens to be contaminated with Praseodymium, of all things? And the contamination is such that the Praseodymium gradually appears to the detector over time, at the same rate that Cesium disappears? And when experiments without Cesium are run, the Praseodymium is mysteriously absent? What a strange coincidence.
Sarcasm aside, though, the experimenters are well aware of this argument, and have a very good explanation for why it couldn't be contamination - namely, isotope ratios. Essentially the distribution of isotope frequencies for the transmuted elements they find are different from the natural isotope frequencies for the same element. Hence, the experiment couldn't have simply been contaminated with the natural version of that element.
The second reason this research counts as strong evidence is that...well, it's actually been replicated. This was particularly bizarre for me to discover upon reading about cold fusion - I was under the impression that there were no clear-cut replications of any cold fusion experiments, anywhere. That's apparently not true though - researchers at Toyota have redone Iwamura's experiment and also find Praseodymium being created. Unfortunately it was presented at a conference, and there doesn't seem to be an associated paper. Here's a link to an article though that describes the replication, though, containing some slides with the Toyota researchers results: http://news.newenergytimes.net/2012/12/06/mitsubishi-reports-toyota-replication/. The article also mentions researchers at two universities (Osaka and Iwate) reporting similar findings.So to sum up: simple elemental detection experiment. Transmuted elements found. Control experiments fail. Multiple confirmations. Combined with the high-quality excess heat measurements of Swartz above, I feel very confident in concluding that cold fusion is a real physical phenomenon. For an additional bit of low-weight evidence, though, I submit to you also the fact that NASA, of all organizations, has an active cold fusion program: see http://futureinnovation.larc.nasa.gov/view/articles/futurism/bushnell/low-energy-nuclear-reactions.html. To be honest I think that article overhypes the current situation; yes, cold fusion appears to be real, but I find the assertion that multiple groups have already achieved kilowatt-level heat production to be very suspect, based on what I've read. Regardless, the fact that NASA is treating this seriously and actively doing cold fusion research might serve as further evidence for skeptical readers.
This concludes my case.
Now, despite the (I think) fairly convincing picture I've painted here, we are still left with the nagging question of why so many early cold fusion experiments failed, and why so many continue to fail today. It seems clear that, real effect or no, cold fusion experiments have unusually low reproducibility. Shouldn't this count against it somehow? In the words of one skeptic, nuclear physicist Richard Garwin,
"It's absurd to claim that experiments that seem to support cold fusion are valid, while those that don't are flawed."
I think Garwin misses the point here, though. What cold fusion advocates are looking for is an existence proof. They just have to show that there exists some set of experimental conditions for which cold fusion occurs. Or, to flip the quantifiers (as PhilGoetz might put it ;), they are trying to disprove the hypothesis that for all sets of experimental conditions, cold fusion never occurs. Looking at it that way, of course a few experiments would be sufficient to make the case - it's just standard Popperian falsification. When you're dealing with "for all" statements, its one strike and you're out.
Or, to put it in Bayesian terms: the probability of getting negative experimental results, conditional on cold fusion being true, is not that low. If cold fusion is true, then somewhere in the experimental parameter space there must be a region where it occurs. But that says nothing about the size of the region; it's fairly easy to imagine experimenters setting out to demonstrate cold fusion and missing some unknown key aspect of the design, giving a negative result. One doesn't even have to posit any experimental error - they're simply looking in the wrong place. On the other hand, the probability of getting positive results in a well-designed, well-controlled experiment, conditional on cold fusion being false, is extremely low. It's basically equal to the probability that the experimenter screwed up the measurement, which can be made vanishingly low with proper controls and replications.
With all that said, of course, it would still be nice to know where exactly previous cold fusion researchers were going wrong. Mitchell Swartz, incidentally, thinks he has this figured out. He's identified a number of necessary conditions for cold fusion that are frequently absent from failed experiments and present in successful ones. The two main culprits seem to insufficient loading of Deuterium in the metal lattice, and a non-optimal (too high or too low) level of electrical driving of the system. I have no idea if he's right about the particulars, of course. But it certainly doesn't seem implausible that this will all be sorted out in the near future, and what seemed like irreproducibility will simply turn out to be the result of an underlying, thus far opaque, pattern.
Huh, this turned out much longer than I expected. I guess I'll close by noting that this topic seems like an almost perfect candidate for confirmation bias; who wouldn't want to believe in a cheap, unlimited, carbon and radiation-free energy source? That's part of the reason I made this post; what I'd really like is for people to a) pick apart this post, looking for flaws in my logic/arguments, and b) look into this whole cold fusion thing independently, and see if they reach the same conclusions. I'm very interested in getting this right, for obvious reasons, and I think at the very least I've made a sufficiently interesting case that doing some research online would be worth it. I don't think I really need to mention the almost mind-boggling impact cold fusion would have, if it turned out to be real and exploitable.
I'm cautiously optimistic about the future right now, LW.
 This is standard history, see http://en.wikipedia.org/wiki/Cold_fusion
Relevant quote: "The explosions are difficult to keep secret. Most people who have been around the field know of them: Fleischmann and Pons in Utah, unidentified researchers at Lawrence Livermore National Laboratory, a group at SRI International, Tadahiko Mizuno in Japan, Jean-Paul Biberian in France, and another situation in a Russian lab a few years ago.
The only lab that may have blown up was the one in Russia. In the other situations, the experiment, not the lab, blew up. SRI International researcher Andy Riley was killed, and Michael McKubre was wounded. Mizuno lost his hearing for a week and came very close to sustaining severe injuries."