All depends on the rat...so long as you're using food as a reward you can teach them to go through mazes and the like, but I seriously doubt mine would pay attention for long enough to learn anything like that. I've also read more than once that they can be taught to respond to their names but mine don't seem to recognise theirs at all. Maybe mine are just thick :-(
About 15 years ago I saw a man on a tube train with a pet rat on his shoulder. It was white, but with pink and blue punk stripes down its back. I don't know whether the man did them, or if the rat did them itself.