At the last gig, bugs went into the ticketing system (CompuWare TrackRecord, a giant turd) like every other issue. This was backed up with the awesome power of whiteboards, emails, and ugly-ass Excel files.
At Worldcom, bugs in telephony things were in a 3270 app called 'CMM', which later got a screen-scraped VB front end, and was eventually replaced with a very bloated Java front end and renamed 'CMMi'. (I'm sure someone thought they were being cute with the names.) Bugs in computery things were tracked in Remedy, and bugs in operational things threw alerts on a very hacked Polycenter Console Manager. Prioritization of those three things went like this: if anything is 'orange' or worse on PCM, do that. Then, either guess, or play email-and-phone-tag with someone who can guess better than you.
Job before that, Phil kept everything in his head.
Job before that, I kept everything in my head or on the whiteboard.