Home of SMF: Free PHP and MySQL forum software
This will do everything you need.
I'm thinking of adding a forum to an established blog site.
What is the best forum script to use which will integrate well with wordpress? I don't make users register to post a comment, so I don't have to worry about existing members.
I have checked out bbpress which is from the same creators as wordpress, but integration looks like a mess.
Any other suggestions? Please don't suggest vBulletin, I know it's a good forum but I don't want to pay for a script considering the forum is not the main part of the site, I just want to get a bit of a discussion happening.
phpbb is also available. but ssmf is better too for it has lots of plugins and mods
I have no doubt it's popular because it's a simple plugin to install. Doesn't mean it's good though
vBulletin is definitely great software - but I think it's too pricey for the OP's project.
bbPress looks good if you can work your way through the integration. Otherwise, I'm still standing behind SMF.
MyBB is also very nice. I would stay away from phpBB though, because you need to manually install all plugins and add-ons.