I'm considering running FogBugz myself, but need to know how much I need to know about SQL Server or MySQL to do so safely.
We like SQL server because it's rock-solid out of the box, and, in our experience almost never corrupts a database. At a bare minimum, you should have someone on staff who knows how to:
One of the best resources on this is MSDN: http://msdn.microsoft.com/en-us/library/ms345583.aspx
We also like Kimberly Tripp's excellent SQL Skills blog.
MySQL Community Edition is free, but free can be very expensive. It can definitely be used as an enterprise-grade database, and we have many people running successfully with databases over 30 GB, but you should have someone on staff who knows how to install, configure, and tune MySQL. (Pointing a Rails app at a MySQL database for a personal project doesn't count. You should have experience with MySQL as a back-end for an application with at least twice the number of users as you have FogBugz licenses.)
You should have frequent backups and the ability to access multiple past backups in case it takes a while for a database corruption to surface.
How much support can I get?
Some. Our support engineers are FogBugz and Kiln experts. We can help you to troubleshoot and administer your database only up to a point. The cost of FogBugz On Demand includes top-notch database administration and backups. If you go with self-hosted, please make sure that you have the right kind of skills to safely and securely administer it.