https://simplemachines.org/

Elegant. Effective. Powerful. Free. SMF is all of the above. SMF is a next-generation community software package and is jam-packed with features, while at the same time having a minimal impact on resources.

https://punbb.informer.com/

PunBB is a fast and lightweight PHP-powered discussion board.

http://phpforum.sourceforge.net/

PHPForum is optimized to provide programmers with a forum who is easy and fast to install. Easy modification is also a feature as all HTML are separated in templates

https://www.phpbb.com/

phpBB is a free flat-forum bulletin board software solution that can be used to stay in touch with a group of people or can power your entire website.

https://www.phorum.org/

https://mybb.com/

MyBB is the free and open source, intuitive, extensible, and incredibly powerful forum software you've been looking for.

https://mylittleforum.net/

https://www.minibb.com/

miniBB® is a standalone, open source program for building your own Internet forum, it's free to download and highly customizable.

http://www.funkboard.co.uk/

FunkBoard, the FREE full function multi-thread, highly customisable, open-source bulletin board messaging system. It uses the powerful PHP language and either MySQL or PostgreSQL database.

http://fudforum.org/forum/