I'm not sure if there is a free forum host without ads, even if you have your own domain. If you have web hosting or your own server already, you can download and install SMF (which is what the 506th uses) or phpBB - i think those are the two most popular free forum softwares.
If you dont care to spend some money monthly and don't want to do your own server management, one of the best paid forums used to be Invision Power's stuff. You can get cloud hosting starting at $30/m and it lets you use your own domain. The other big name in paid forums is vBulletin, who also does cloud hosting at $30.
None of these forum cloud options will probably have a static front end website either, though the forum software itself may have addons or similar Content Management System functionality.
If you want to self host, it will only cost a smaller price with any web host, but you will have to self maintain - updates, backups, etc. (Edit: Correction, updates and backups of the forum and maybe database will probably be required. A standard shared web host wont have you do server updates.)
Hit me up if you have more questions or need help setting up a self host option, been there done that.
