mateuszbb is a minimalist, easy-to-use, easy-to-administer and distraction-free forum engine written in PHP. MIT-licensed.
https://sourceforge.net/p/mateuszbb/code/HEAD/tree/
or
svn checkout svn://svn.code.sf.net/p/mateuszbb/code/trunk mateuszbb
Here below are a few examples of mateuszbb being used in real situations. Let me know if you use it on your website, I will be happy to add a link to your forum as well.