I love finding mistakes and continuity problems, but I usually point them out as I'm watching a movie. It's too hard for me to recall one off the top of my head. Ravs already mentioned the only big blooper I tend to remember: Indy's snake in the glass.
The easiest mistakes to catch are in reflections. This doesn't occur quite so much now, as CGI can forgive a multitude of sins. But still, when a character is wearing glasses or sunglasses, sometimes you can see the set and the whole crew reflected in their glasses.
I also like watching keeping my eye out for the boom in the shot, or a "hair in the trap."