If there's one thing I can't stand, it's willful incompetence (there are actually more things I can't stand, as you probably suspected). If you are doing a job, even doing it half-assed, you should do it right, even if you're just doing the minimums. Don't screw things up for everyone else. If asked to perform a task, do it. If you don't want to go above and beyond the minimum acceptable requirement for the task, don't, but don't willfully screw it up or ignore it.
I work with people who think that the best method to get through the day is to do as little work as possible and kiss enough ass to make it look good. Unfortunately, this works well enough. Meanwhile, others are busting their asses and get jumped on for a simple mistake.
Our office was asked to provide a shuttle service to and from two off-campus locations. One of my coworkers remarked that he wouldn't drive it because "I don't get paid enough to do that. I pretty much just get paid enough to sit around and play video games." He further commented that he would refuse if directly asked.
This same employee will then come up with some sort of "helpful suggestion" to make the office run more smoothly. These ideas conveniently spring up when the boss is around, and he never actually follows through. On the other hand, several employees have fixed problems themselves whenever they saw a way to make things work better.
He also leaves whenever his relief shows up, be it 5 minutes before quitting time or 45. He does this without apprising the relief of the current status of the various buildings on campus, saying so much as a "thank you" for getting there early (something he never does), or really doing anything other than bitching about having to work.
If the patrol car needs to be fueled at the beginning of his shift, you can be damn sure his relief is going to need to fill it, since he won't.
I know that every workplace has at least one of this guy, but it really ticks me off when the boss doesn't realize or seem to care. But I guess I should just do my job and let everyone else worry about doing theirs, even if I know they won't.
Another sort of willful incompetence is found in the Assistant Director of our little office. She doesn't answer her phone, takes off early and arrives late, parks in the fire lane, and reams anyone who does something differently than she'd like it done. She does things wrong just because she doesn't like the boss, and she's been trying to get me fired ever since I made it clear that I don't put up with crap.
At least I kind of see why she's that way, though. She's got some problems she needs to deal with. The other guy, though, is just a dick.