The main one nowadays that doesn't feel like something from 2002 is Discourse. Its free, open source and has a pretty good community and support.
I know it supports markdown and I'm fairy sure it has BBcode support too (though I would highly suggest you use markdown instead). Discourse does run on ruby though, which may not perfect for everyone
I will highly recommend vBulletin. It is a great platform and comes with smart and easy customizations. Additionally, it comes with a lot of customizable design templates for users to choose from. There are a few others in the market like phpbb, Simple Machines Forum and more. You can check those too, to see which suits you better.
There are lot of traditional forum softwares, but among all of them, I would choose the three most reliable: phpBB, vBulletin, and Discourse. Its simple to use and has a neat and simple design. Finally it all depends on your need and budget, you can validate these softwares and pick one that suits.
MyBB is a free, open source and powerful forum software written in PHP language and uses MariaDB to store their data. It provides lots of features such as support of various plugins, widget support, customizable themes and many more. MyBB have very very large plugin support list.
I visit forums very often and I notice that most of the highly functional forums are using vBulletin.
There are other forum software available as well.
- phpBB (free, open source)
- simple machines forum
SMF (simple machine forum) if you want free (support is not guaranteed, a lot of mods and themes for free)
vBulletin if you want paid (good support, mods and themes paid/free)
using both of them and they are really good