TITLE: How to pick a thesis advisor?
QUESTION [38 upvotes]: This sort of question is probably in bad taste for math.stackexchange, but is probably in high demand. (I tried to start a site on Area 51 to house questions like this, but my request was closed due to the existence of math.stackexchange.)
I am advising talented students about graduate school. I believe that the most important thing one must be sure to do is pick the right PhD thesis advisor.

Question: What is your best advice on picking a good thesis advisor. (For the sake of levity, feel free to answer an opposite question regarding how to pick a bad thesis advisor. Just be sure you are clear whether you are indicating how to pick a good or bad advisor!)

REPLY [3 votes]: When sizing up a potential advisor it's helpful to chat with their current students, especially those nearing graduation.  How much help do they get, both in choosing problems to work on and in working on them?  How do they balance their desire for you to improve your thesis with your need to graduate in a timely manner?  How likely are they to make the effort to help you with your career (placement, tenure review) after you get your degree?
I could rattle off quite a few tragic cases, where extremely talented mathematicians of my generation were irrevocably derailed and are "flipping burgers" due to either failing to ask those questions or ignoring the answers they got.