I am finishing my Ph.D. and I am waiting to hear back from postdocs. I don't have offers yet (although I heard that some schools have already offered) but I have a question about how to tell which school is the best fit in terms of how to be in a better position when I'll be applying for tenure track positions. I want to hear about what kinds of criteria were used when people were choosing postdocs.

I think that it is usually good to choose better-known institutions and larger institutions because that helps you not fall through the cracks when you apply for tenure-track jobs. But you also want someone whose research interests align well, and someone who is well-established such that they can write you a good rec letters when you apply for tenure-track jobs. Combine that with your personal constraints (two-body problems etc) and what you get might even be an empty list of schools.

How did you prioritize among all the different constraints for ranking postdoc institutions? Is there anything that you wish that you knew when you were deciding where to go?

P.S. Pretty sure that this is a community wiki question, but this is my first time posting on mathoverflow and I don't know how to make this a community wiki.