I have several pet peeves, but I'll list a couple here.
1. Seeing people load up all sorts of unnecessary gear on their horse in the name of "training" - especially when its being used incorrectly and possibly ruining what could have been a good horse. Its even more frustrating when they claim its their trainer doing it and don't have enough common sense to question their trainer about it or look for a new trainer.
2. Getting a bigger bit in a situation where it really isn't the solution.
3. People claiming their horses are out of big-name lines when that big name so far back its off the papers.
4. People claiming the improper height on their horse (saying their horse is 17 hands when its obvious its more like 15) and not knowing how to denote hands - 16.5 hh instead of 16.2 hh.
5. People who want to breed but don't have the slightest understanding on proper conformation or what makes a good breeding animal.
And, stallion owners acting like fools online. I've seen more than one big name stud owner acting like a tart online bashing other stallions..