At this point phpBB is probably the safe bet, but I would like to put in a plug for SimpleMachines Forum (simplemachines.org). It's in late beta, so it's not a "right now" option, but it's also not "just another half-finished forum project"--it's actually a ground-up rewrite of YaBBSE, which is in turn a php/MySQL-based forum loosely based on the old YaBB Perl forum script.
So, there's a lot of experience behind the project, and in my opinion they did a very good job through the last revision of YaBBSE, and the SMF project has been moving along steadily. It's not open source, which would be its only disadvantage, but the dev team does appear to work hard.
I ended up going that route because I got into YaBB early on, and it was easier to go to YaBBSE than phpBB (and at the time I didn't like the phpBB interface that much, though it's gotten a lot nicer since). I don't feel like I made a mistake, though, and I'm really liking where SimpleMachines is going.