Continuity mistake: When the guy in suspenders sneaks a cassette tape of Harry's show into the boombox, at the place where the students are dancing, the recording of Harry busting Deever is a completely different performance. You can tell because the word "Why" (from "W-w-why is that, Mr. Deever?") is much lower in pitch.
Continuity mistake: When Mazzilli is talking to the reporter in front of the school, he lights smoking firecrackers and throws them behind him. When we see him light the first one, it is at least three or four inches long when we see it in his hand, but much shorter an instant later when it hits the ground. This happens again with the next one too, but this time it's shorter when we first see it and slightly longer when it hits the ground.
Continuity mistake: In the scene where Christian Slater's character is talking to "Serious" (the kid who commits suicide), the letter he received from him is handwritten on lined paper. Earlier in the movie, there is a scene showing "serious" typing the letter on his computer.
Continuity mistake: When Paige is shown brushing her hair in front of the mirror, there is no background shown in the mirror behind her; it's completely black. We see this a few times while she's sitting at the dressing table, but then when she gets up to gather her things and take them to the kitchen, we see the room reflected normally.