The only major problem we have experienced was once on the Tower of Terror when, after the drops, the car stopped at the bottom and quit moving. The car was dark, and it wasn't moving at all. Sat there for about five minutes without a sound from outside. It seemed much longer than five minutes. I had ridden the attraction several times in the past, and I knew that a door was supposed to open and let us out. I kept wondering if a second car might fall on top of ours. (Probably not possible, but the imagination is strong when one has been in a dark car after a 13 floor drop.)
Then, behind us, a young girl started screaming "I don't want to die!" She was really upset, and everyone else was getting quite upset as well.
Finally, the door opened and let us out. The CM just stood there like everthing was normal, not saying a word, not showing any emotion at all.
My wife has refused to ride Tower of Terror since, and I've been a bit apprehensive about the ride myself but still ride it.
As far as common breakdowns, the two rides that seem to me to break down the most often are the Haunted Mansion and Test Track. But I did sit in my log for about fifteen minutes in the middle of Splash Mountain once trying to hear the announcements that could not be heard because of the loud music which never stopped.
Enjoy.