My first instinct was to use the Track Issues module of my ERM (implementation of which has proven to be klunky beyond my wildest imagination, but that's a different story). The back end interface was cumbersome and provided no way (at least not that I could see) to add updates to individual issues/problems. Also, it would have required the activation of the staff mode, which is not currently used and would most likely not have been embraced by users.
Fortunately, we are using Drupal for our new intranet site and it comes with a support ticket module. It's beautiful. It may not use terminology I would have picked (for example, client area for department), but I was able to basically figure this module out in less than a day. I tested it for a couple of weeks and officially rolled it out for use yesterday. We customized nothing.
I've implemented software in a previous life and understand the ramifications of over customizing applications. This module does the job. Maybe we would have changed a few words here and there, but in the end I didn't think it was worth it. It isn't always necessary to customize something. If it works, just use it and move on.