Should I mention my Stack Overflow (and other Stack Exchange sites) reputation in my CV while applying for a post-graduate position?
No, you shouldn't. Not yet, anyway
In general, I think it's perfectly fine to list that information on a CV for an academic position. Depending on the profile of the position itself, I would feature it more or less prominently. Say, if you apply for a scientific programming position (or a position with heavy coding), you could list it under a “Skills” section where you would say:
C/C++/Fortran with a focus on shared-memory and distributed memory parallelism (OpenMP/MPI)
Received formal training at XXX National Lab, taught parallelism course at University of YYY, involved in StackOverflow (username: zzz) on this topic.
(if the format allows it, like a PDF, consider including a hyperlink)
If the position is not one heavily involving code-writing, you could tone it down, or even list it in a “Hobbies” or “Personal” section. Many people like to list hiking or book reading or civil war reënactments, so why not list Stack Overflow!
But… in all cases, only do it if your account is of the wow! type. You don't have to be Jon Skeet (it may take years of therapy to accept that, but that's the sad truth), but you don't want someone to look up your profile and think “meh”.
I would say not to. As a hiring manager I really care most about the relevant details such as work experience academic focus. Although SE/SO is pretty darn popular it's also a website and, although we may disagree about this :-), not seen as a professional/peer-reviewed/authoritative/juried/etc. source. I would just see it at fluff and wonder why it was there. If you're going for a research position keep in mind that these people haven't seen sunshine in months, much less a computer that doesn't have Matlab open on it.
I would, however, be interested if somebody explained to me (in the interview) what this whole thing was when I asked the "so what else are you into" question. As a hobby this shows that you are a technologist first and that you make your geek part of your life. I would take that into consideration for an academic or a professional position because it shows where your interests are.
In general; I would say that if you feel strongly about something- don't put it on your resume. Save that for the interview and make a good impression with it.
EDIT: Nate Eldredge made a good comment below. Although it should be obvious, it is worth pointing out that this post is made from an industry perspective. It is provided to frame the topic within the larger context of an interview; any interview. My experience has been that there is very little difference between my professional and academic interviews, ymmv - of course, and therefore I submit that the commentary is germane.
Re-reading my comments I realize that, yes, I was being a bit flippant for comedic effect. No offense intended.
If I was to hire a Postdoc, Phd student, a Master student or a programmer for my project and that position had a related component (lets say Programming, GIS, Maths, CS, Mathematica etc.) and someone had a very good reputation in the site that I understood, I will definitely see it as a strong indicator. Of course it's easy to check the type of questions that have been asked/answered and the calibre of the person => if someone develops a strategy to just get points by answering easy questions and asking general questions that are bound to get a lot of up votes that would not win a lot of brownie points. Nevertheless I will never penalize anyone for it. It's active participation which is always positive.
I can't think of a situation where it would be helpful to list SE reputation on your CV. Most people won't know what SE is, and so will either not care or think it's weird. If your reputation is not very high then it's also going to look bad. Finally in the one situation where it might help you (the person reading your CV is active on SE and you are a SE superstar), there's no point in listing it because the person will already know who you are. I don't need a CV to tell me that Qiaochu Yuan is a major contributor to MO.
It depends on the field. Unless you're sure that it's well-known and valued in your field, don't write it as a claim of competence — but you can write it as a hobby.
Let's not over-estimate the value of Stack Exchange. Stack Overflow, by far the largest website, has an Alexa ranking of 86. It's probably well-known among programmers, but probably not among others. Stackexchange.com in its entirety has a ranking of 469. That includes Stats, GIS, Maths, CS...
superuser.com has a ranking of 1620,
serverfault.com a ranking of 2159.
Most likely, an academic reading your application will not know the website you are mentioning. A CV should focus on the important parts. Any space wasted is harmful to your cause of grabbing their attention. Mentioning a high score on a website they haven't heard about is a waste of space. Therefore, I would not put it on your CV, unless you're sure it's going to impress them.
That being said, I've heard of people writing in their CV that they're among the top 200 in World of Warcraft. It's perfectly fine to write hobbies in the CV — but if you're trying to convince someone of your competence by citing your score on a website they're not familiar with, you might do more harm than good.
I don't think it is a good idea because the value of a "SO rep point" is unfamiliar to most of your audience (and varies across sites). Still, SO provides a useful public record of thinking, learning, teaching, dialogue, and social skills.
One way to leverage this would be to point to examples of your teaching, learning, and problem solving - with perhaps a well chosen example in the research and / or teaching statements; if your published results benefited from SO, a reference to a SO question could tell an interesting back-story behind a paper, and be a launching point for anyone who is curious about your other contributions.