Invision and WordPress
Invision has decreased in popularity in recent years, but is still one of the big contendors. Meshing it with WordPress is pretty simple and the best option I could find was InvisionBridge WordPress-IPB Bridge (http://forums.invisionize.com/lofiversion/index.php/t125596.html) which combines the login system for each and also offers some extra options in the paid version (which is $30), including posting topics and comments on WP as threads and replies in IPB and vice versa, avatars and PM integration and a few others.
Integrate SMF and WordPress
SMF is fast growing in popularity and is a personal favorite of mine. It’s fast, reliable and does everything that phpBB does but better. It also lends itself to other software very well and so hooking it up with WordPress is a sinch.
All you need to do is download the WordPress SMF Bridge plugin (available fromhttp://www.earthorbit.com/opensource). It installs like any other plugin and means that when users sign into your forums they’ll also sign into WordPress and vice versa. You can also set it to post treads on SMF as topics on WordPress, display your SMF stats and a few other things.
Alternatively you can try the original upon which the above version was based at http://www.dmry.net/wordpress-smf-bridge-plugin-10/
Integrate vBulletin and WordPress
vBulletin is one of the more professional forum software packages and while phpBB is usually seen on sites which get 5 visitors per day, 4 of which are the web-masters friends and family, vBulletin is seen more on serious sites where SEO and design matter.
Since vBulletin costs money to buy and use, it has an excellent team behind it and support and great functionality. To bridge vBulletin with WordPress is an extremely simple process and can be done by downloading this vBulletin plugin (Complete WordPress/Vbulletin Bridge – Share Users And Postings - http://www.vbulletin.org/forum/showthread.php?t=134521) and following the instructions.
Integrate WordPress with phpBB
phpBB is the big daddy, the long standing heavyweight of the forum packages. This is mainly due to the fact that it was the first major open source forum solution and was free from the get go. phpBB has more plugins, mods and extensions than just about any other piece of software, and so integrating it with WordPress is again, pretty straight forward.
The best options I could find are WP-PHPBB WordPress plugin (http://www.avalon5.com/wordpress/themes-plugins/wp-phpbb-wordpress-plugin/) which installs just like any other WordPress plugin and works well. The only requirement is that your WordPress and phpBB installations are using the same database, though if you’re handly with PHP you could probably get around this.
Alternatively there’s the more up to date WP-United (http://www.wp-united.com) which has a bigger ranger of options and attributes including login integration, giving blogs to your forum members, easy setup, theme integration and more. For all the forums, this is the best package I’ve seen out there and both the forum system and plugin are absolutely free.
While some of these options may take a little playing around with before they’re functional, most work out of the box. One problem though is that when you update WordPress to a new version, you may be left with forums that don’t work or don’t integrate. If this is a problem for you I recommend checking out The Top 4 WordPress Forums Plugins