Choosing a postdoctoral position

I had a request a while ago for a post about how to choose a postdoctoral position (from the point of view of a finishing-up grad student, I'm assuming).  This is a tricky topic, precisely because it's somewhere between choosing a grad school (lots of good places to go, with guaranteed open positions every year) and getting a faculty job (many fewer open positions per year in a given field, and therefore a much restricted field of play; plus, a critical need to make some hard decisions that could be postponed or avoided in grad school).  Moreover, different disciplines within the physical sciences have very different approaches on postdocs.  In some fields like astronomy, externally funded fellowships sponsored by observatories/facilities/programs are standard practice, while condensed matter physics is much more principal-investigator-driven.  So, I'll try to stick to general points.

• I strongly suggest going somewhere that is not your graduate institution, unless there are strong extenuating circumstances.  It's just intellectually healthier to get a broad exposure to what is out there, rather than to stay entirely comfortable.
• This is also one of the relatively few points in your career when you can really shift gears, if you are so motivated.  My doctorate was in ultralow temperature physics, but I decided to become a nano researcher, for example.  More dramatically, this is often the point where many people get into interdisciplinary fields like biophysics.  There are trade-offs, of course.  If you do a postdoc in an area very close to your thesis work, you can often make rapid progress.  On the other hand, most people who go on in research (industrial or academic) do not end up working on their thesis topic for the lion's share of their career, and this is a chance to broaden your skill set and knowledge base.
• Word of mouth and self-motivation are essential to getting a good postdoc position, beyond posted ads.  If you're finishing up in grad school, you are enough of a professional that you should be able to email or otherwise contact people whose work you find interesting and exciting, and ask whether they have any postdoctoral openings.  You should make sure that these emails are reasonably detailed and that it's clear they're personalized - not a form letter being spammed to several hundred generic faculty members simultaneously.  Your hit rate won't be high, but it's better than nothing.
• Don't discount industry, though it's a narrowing field.  There are still industrial postdoc positions, and if you've got an interest in industry more so than academia, then you should look at these possibilities.  This includes places like Bell Labs (yes, they still exist), IBM, Intel, HP Labs, etc.  It is a tragedy that there aren't more opportunities like this out there now.
• You need to think about how a particular postdoc position is structured.  Are you going to be acting as middle-management, helping to mentor a team of grad and undergrad students?  Are you going to be leading a research project yourself?  Is there a lot of lab-building or lab-moving?  How long is the position, and how does it match up w/ the seasonal nature of academic hiring, if academia is what you want to do?  Where have previous postdocs in that lab or group ended up?
• How set are you on academia?  If you are set on academia, what kind of academic position would make you happy?  Go into the academic track with your eyes open!  If you're looking beyond academia, what do you need out of a postdoc position (besides a paycheck)?  Are there particular skills you want to learn?

None of this is particularly insightful, but it doesn't hurt to have this written down in one place.  Suggestions for further things to consider are invited in the comments....