Deliberate mistake: At graduation, after Troy is handed the diploma, in the overhead shot the graduates on the left and right stand-up from their seats, then turn and walk towards each other. All their seats simply disappear between shots. Obviously done deliberately for the dance sequence, but a mistake nonetheless.
Deliberate mistake: Several times, you see an exterior shot of Nathan's house, and it is very clear that it is the same shot being used over and over, sometimes reversed.