Free: phpBB just always was easy to custom mod.
Paid: IPB but I think XenForo is making some nice gains.
Free: MyBB, SMF and phpBB
Paid: IPB
Yea I know I have more than one choice for free but those really are very good choices. And IPB as my paid choice since I have been using it ever since my first registration to a forum. I also tried VB but I never liked their ACP.
Free: MyBB and PhpBB
Paid: IPB and vBulletin
Free: phpBB
Paid: IP.board
I've never personally administered a forum running myBB, so I can't say anything about that.