Plot hole: When the first police car arrives at the observatory, the policeman there tells them that the kid (Plato) holed up in the observatory has a gun and had "wounded a kid earlier." However, when that same policeman arrived at the mansion, he and his partner only ran into Plato and then followed him to the observatory. He had absolutely no way of knowing what had happened at the mansion prior to them getting there, which is when Plato shot one of Buzz's buddies.
Continuity mistake: At the beginning of the movie at the police station and the next morning at breakfast, it at least appears to be established that Grandma lives with Jim and his parents. After all, they had just moved from another city and Grandma was there early in the morning at breakfast time. However that's the last we see of her. She is never mentioned again nor is she there after Jim comes home from school or at anytime after the trouble starts. It would be unusual for someone to sleep through all that went on; but no one ever says anything about "being quite or we might wake Grandma up." After that scene at breakfast, she is forgotten.
Trivia: At the very end of the film the camera is zooming out with the observatory in the middle of the screen. Right before the credits begin, a man in a brown suit carrying a briefcase walks up to the observatory. This man is Nicholas Ray, the director of Rebel Without A Cause.