We use Request Tracker for both external support and internal projects, are extremely happy with it. It is extremely customizable, and our customers like the login system. In fact, several of our customers have asked us to install extra instances for them to manage their own support requests (So far one of our hosting resellers, a school IT department, and a real estate agency)
It especially strikes a good balance between imposing some structure, and yet not being so rigid that it doesn't work in the real world.
The negative side is that it takes a good deal of effort to get it installed and configured.
Good luck finding something that works for you.
Regards,
Erek Dyskant