What is the Best Forum Solution for WordPress?

This will depend on your circumstances and requirements but bbpress is by far the most impressive for wordpress. It comes with an import tool that allows you to import your posts from 20 other forums. There is also a huge amount of extensions for it. There are others that are also a good match like: the free version of WP Symposium which is a social network/forum combination. Uses shortcode to implement into your site and is very user friendly. CM Answers is another one which offers a free and pro version and is more of a question and answer format. The free version is somewhat under powered compared to bbpress and wp symposium.

An external solution will work but will take more effort to make it look and feel like your existing wordpress site. And if you want to bridge the login process with your wordpress site you may have to hire someone to create the bridge script for you or pay for one already in the market. I have set up bbpress personally for clients and prefer it to the others as I stated it has a lot more extensions and support. Good Luck and hope this helps.

Or you could get a forum with a CMS aspect to it. It depends on if you want a forums site with blog, or a bloggy site with a forum.

It depends how big you plan your forum to grow.

I have used Mingle in the past and still use it to this day. It's perfect for customer support type forums but I wouldn't use it for very large forums. If I were building a large forum then I would use something like vbulletin. You can still integrate it with Wordpress using a membership script such as Digital Access Pass.