The Webmasters FAQ outlines the nature of acceptable questions for this site: researching available solutions and providing anecdotal advice falls outside the purpose of this StackExchange site, so this type of question would be better-suited to chat or another discussion forum.
That being said, there are a variety of free and open source content management systems, as well as licensed software and paid services, which may meet your requirements.
Some of the most popular forum scripts include:
- PHPBB (php, free and open source)
- Pros: been around for a long time, so there are lots of documentation and mods for it. Has lots of features.
- Cons: installing mods is not easy
- Fluxbb (php, free and open source)
- Pros: very lightweight and simple
- Cons: hard to install mods, not a lot of features.
- vBulletin (php, pricing varies by suite)
- Pros: very robust and powerful, widely used
- Cons: very expensive
- Vanilla Forums (php, free)
- Pros: easy to use, with a great user interface
- Cons: isn't like a traditional forum, and some of the features, like badges, aren't free.
- BBPress (php, free and open source)
- Pros: plugin version integrates really well with wordpress, simple, seo-friendly urls built in, easy to modify and hack if you know wordpress.
- Cons: often need to integrate bbpress with your wordpress theme.
- SMF (php, free and open source)
- Pros: great mods, and a great community
- Cons: can be hard to customize if you aren't used to the admin interface
- MyBB (php, free and open source)
- Pros: tons of features
- Cons: amount of features can be overwhelming
- XenForo (php, commercial)
- Pros: lots of great features, such as badges.
- Cons: it's not free.
All of the forum software listed above are built around the concept of mods (plugins) which extend the application's functionality - there are many mods, with new mods authored every day.
Sites like forum-software.org can help you compare different forum software, and have forum comparison tools and in-depth reviews.
If you have researched different forum software and available mods but cannot find anything that will meet your needs, you should consult a developer (or do it yourself) who can build something to your specifications either as a mod for an existing forum script or as a stand-alone application.