There are a lot of bad arguments out there. Fortunately, there are also plenty of people who stand up against these arguments, which is good.
However, there is a pattern I observe quite often in such counter-arguments, which, while strictly logically valid, can become problematic later. It involves fixing all of one's counter-arguments on countering one, and only one, of the original arguer's points. I suspect that this tendency can, at best, weaken one's argument, and, at worst, allow oneself to believe things one has no intention of believing.
Let's assume, without much loss of generality, that the Wrong Argument can be expressed in the following form:
A: Some statement.
B: Some other statement.
A & B -> C: A logical inference, which, from the way B is constructed, is a fairly obvious tautology.
C: The conclusion.
Unfortunately, most of the arguments I could choose for this discussion are either highly trivial or highly controversial. I'll choose one that I hope won't cause too much trouble. Bear in mind that this is the Wrong Argument, the thing that the counter-arguer, the person presenting the good, rational refutation, is trying to demonstrate to be false. Let's designate this rational arguer as RA. The person presenting the Wrong Argument will be designated WA (Wrong Arguer).
WA: "Men have better technical abilities than women, so they should get paid more for the same engineering jobs."
WA relates a terrible sentiment, yet a pervasive one. I don't know anyone who actually espouses it in my workplace, but it was certainly commonplace not so long ago (musical evidence). Let's hope that RA has something persuasive to say against it.
Based on what I've seen of gender discussions on other forums, here's the most likely response I'd expect from RA:
RA: Don't be ridiculous! Men and women are just as well suited to technical careers as each other!
... and that's usually as far as it goes. Now, RA is right, as far as anyone knows (IANAPsychologist, though).
However, WA's argument can be broken down into the following steps:
A: Men, on average, have better technical skills than women.
B: If members of one group, on average, are better at a task than members of another group, then members of that first group should be paid more than members of the second group for performing the same work.
C: Men should be paid more than women for the same work in technical fields such as engineering.
Trivially, A & B -> C. Thus RA only needs to disprove A or B in order to break the argument. (Yes, ~A doesn't imply ~C, but WA will have a hard time proving C without A.) Both A and B are unpleasant statements that decent, rational people should probably disagree with, and C is definitely problematic.
So RA sets about attacking A. He starts by simply stating that men and women have equal potential for technical talent, on average. If WA doesn't believe that, then RA presents anecdotal evidence, then starts digging up psychological studies. Every rational discourse weapon at RA's disposal may be deployed to show that A is false. Maybe WA will be convinced, maybe he won't.
But what about B? RA has ignored B entirely in his attack on A. Now, from a strictly logical point of view, RA doesn't need to do anything with B - if he disproves A, then he disproves A & B. Attacking A doesn't mean that he accepts B as true...
... except that it kind of does.
What if WA manages to win the argument over A, by whatever means? What if WA turns out to be an evolutionary psychology clever arguer, with several papers worth of "evidence" that "proves" that men have better technical skills than women? RA might simply not have the skills or resources to refute WA's points, leading to the following exchange:
WA: Men are better engineers than women, and should be paid more!
RA: That's ridiculous. Men and women have identical potentials for technical skill!
WA: No they don't! Here are ten volumes' worth of papers proving me right!
RA: Well, gee, who am I to argue with psychology journals? I guess you're right.
WA: Glad we agree. I'll go talk to the CTO about Wanda's pay cut, shall I?
RA: Hang on a minute! Even if men are better engineers than women, that's no reason for pay inequity! Equal work for equal pay is the only fair way. If men really are better, they'll get raises and promotions on their own merit, not merely by virtue of being male.
WA: What? I spent hours getting those references together, and now you've moved the goalposts on me! I thought you weren't meant to do that!
RA: But... it's true...
WA: I think you've just taken your conclusion, "Men and women should get equal pay for the same work", and figured out a line of reasoning that gets you there. What are you, some kind of clever arguer for female engineers? Wait, isn't your mother an engineer too?
Nobody wants to be in this situation. RA really has moved the goalposts on WA, which is one of those Dark Arts that we're not supposed to employ, even unintentionally.
The problem goes deeper than simply violating good debating etiquette, though. If this debate is happening in public, then onlookers might get the impression that RA supports B. It will then be more difficult for RA to argue against B in later arguments, especially ones of the form D & B, where D is actually true. (For example, D might be "Old engineers have better technical skills than younger engineers", which is true-ish because of the benefits of long experience in an industry, but it still shouldn't mean that old engineers automatically deserve higher pay for the same work.)
Furthermore, and again IANAP, but it seems possible to me that if RA keeps arguing against A and ignoring B, he might actually start believing B. Alternatively, he might not specifically believe B, but he might stop thinking about B at all, and start ignoring the B step in his own reasoning and other people's.
So, the way to avoid all of this, is to raise all of your objections simultaneously, thusly:
WA: Men are better engineers than women, and should be paid more!
RA: Woah. Okay, first? There's no evidence to suggest that that's actually true. But secondly, even pretending for the moment that it were true, that would be no excuse for paying women less for the same work.
WA: Oh. Um. I'm pretty confident about that first point, but I never actually thought I'd have to defend the other bit. I'll go away now.
That's a best-case scenario, but it does avoid the problems above.
This post has already turned out longer than I intended, so I'll end it here. The last point I wanted to raise, though, is that an awful lot of Wrong Arguments (or good arguments, for that matter) take a form where A is an assertion of fact ("men are better engineers than women"), and B is an expression of morality ("... and therefore they should get paid more"). There are some important implications to this, for which I have a number of examples to present if people are interested.
To summarise: If someone says "A and B are true!", don't just say "A isn't true!". Say "A isn't true, and even if it were, B isn't true either!". Otherwise people might think you believe B, and they might even be right.