I currently use vBulletin, IPB and MyBB. Out of the three I like the vB 3.X series and the latest IPB. I'm just launching communities with MyBB so I don't have enough hours working with the code yet to have an opinion.
If MyBB gets Facebook Connect and Twitter Connect, then it will win for sure. Nonetheless its simplistic Admin CP IPB 3 is also a good choice. It's got excellent SEO out of the box along with Facebook/Twitter Connect as well.
I kind of find phpBB is not modern enough - it lacks AJAX features (ie. you don't need a second reload for quick reply) and a proper plugin system that does not need re-edits after upgrades. I have a problem with vBulletin more though, as Internet Brands really pushed it with their price increasements, bloatware (in vBulletin 4), and its ridiculous TOS (they want your ad info).
After working with XenForo for some time, I really like it over vBulletin. It is not as feature-rich as vBulletin but for anybody looking for a really nice 'down to basics' forum package then XenForo is worth considering.
For free packages, I would likely have to recommend MyBB at this point. The support forums are bit iffy if you need help but for a free open-source project it is pretty good.