Until recently I would have recommended
phpbb forums, as I have a few of those in use.. but... I have been working with
Invision Board lately, and I think its great! You can pretty well build them however you want. User interface is very easy to use. I also find Invision a fair bit faster too. You can see an example of Invision Board
HERE .
You'll find there are a lot of hosts out there that support php/mysql. The ones that arent are slightly archaic IMO, as so many sites are using php now.