Technologal weirdtopia:
95% of all innovators have joined one of three factions: Cyberpunk, Biopunk or Steampunk, whose conflicting aesthetics make the Mac/PC debates look like playground scuffles.
As soon as one faction makes a new gadget, the others stubbornly construct their own version based on THEIR tech. (The biological toaster must be seen to be believed, and the continued existence of the Steampunk pneumatic-tube-based Internet is an ongoing miracle.)
Most "Square" technologies have been appropriated by one faction or the other; for example 'zipper' technology is officially classified as Steampunk, 'velcro' is Biopunk, and the easily-fabbed zip-ties are of course Cyberpunk.
A coalition from the remaining 5% of innovators recently TRIED to break out of the mold and form a workshop for neglected Square technologies such as fusion, meta-materials and superconductors, but now everybody else is referring to them as "those Crystalpunk guys" and several of the contributing scientists have recently begun sporting a crystals-and-togas motif...
Sexual weirdtopia:
Sens-o-tape technology can record and play full sensory experiences, and as usual, the porn industry is the first to cash in on the new trend. Naturally, 'amateur' sex pales in comparison to commercially produced, computer-enhanced experiences, and as a result, real-life sex dwindles in popularity, mostly performed by highly trained professionals on each other, or by long-term couples who have had the time to work out each other's specific buttons.
Economic Weirdtopia:
After population pressure have shifted the economic basis for money to "reproductive success", having ten kids is now restricted to billionaires.
Harsher inheritance laws and gift laws have not only abolished unequal inheritance, but the money is also 'cashed in', giving heirs an non-refundable increase in allowed children. As a result, the measure of 'real' wealth becomes having a diverse network of affluent relatives to support your commercial endeavors via goods and services.
After a hundred years of this policy, the top 1% have become the top 10% - however, most of those people actually belong to sprawling interconnected clans.
On the bright side, the middle class have been substantially bolstered, and as parenthood shifts to becoming the prime status symbol in society, increasing amounts of time and resources are spent on education and child health.
Governmental Weirdtopia:
Every political candidate who gets even a single vote is elected to office. ("No votes wasted!") Consequently, there are over 5000 presidents of the United States (It would be more, but those with a constituency of less than 1000 people tend to drop out because the hassle isn't worth it.) Some presidents are simultaneously staff members of other presidents. Many presidents elected other people to be THEIR presidents. All presidents dictate policy, which affects precisely those citizens who voted for them.
A special network of supercomputers is continually busy calculating what laws, regulations and taxes apply to each particular citizen.
Cognitive Weirdtopia:
After the secrets of the mind have been unlocked, personality traits can now be bought and sold on the open market. Everyone wants a piece of successful businessmen's brains, and many people pay to have the currently-trending virtues and vices installed. There is a popular movement to mandate Integrity, if not Honesty, in anyone serving as a politician or CEO. (This is heavily resisted by current politicians and CEOs, but the install seems to be a competitive advantage for the up-and-coming politician.)
Utopia and Dystopia have something in common: they both confirm the moral sensibilities you started with. Whether the world is a libertarian utopia of the non-initiation of violence and everyone free to start their own business, or a hellish dystopia of government regulation and intrusion—you might like to find yourself in the first, and hate to find yourself in the second; but either way you nod and say, "Guess I was right all along."
So as an exercise in creativity, try writing them down side by side: Utopia, Dystopia, and Weirdtopia. The zig, the zag and the zog.
I'll start off with a worked example for public understanding of science:
Disclaimer 1: Not every sensibility we have is necessarily wrong. Originality is a goal of literature, not science; sometimes it's better to be right than to be new. But there are also such things as cached thoughts. At least in my own case, it turned out that trying to invent a world that went outside my pre-existing sensibilities, did me a world of good.
Disclaimer 2: This method is not universal: Not all interesting ideas fit this mold, and not all ideas that fit this mold are good ones. Still, it seems like an interesting technique.
If you're trying to write science fiction (where originality is a legitimate goal), then you can write down anything nonobvious for Weirdtopia, and you're done.
If you're trying to do Fun Theory, you have to come up with a Weirdtopia that's at least arguably-better than Utopia. This is harder but also directs you to more interesting regions of the answer space.
If you can make all your answers coherent with each other, you'll have quite a story setting on your hands. (Hope you know how to handle characterization, dialogue, description, conflict, and all that other stuff.)
Here's some partially completed challenges, where I wrote down a Utopia and a Dystopia (according to the moral sensibilities I started with before I did this exercise), but inventing a (better) Weirdtopia is left to the reader.
Economic...
Sexual...
Governmental...
Technological...
Cognitive...