Deliberate mistake: During almost every scene that involves the main actors in the hallway or in a classroom with a window to the hallway, there are people passing by at random intervals. Even if the scene takes place in the middle of class or after school, there are random extras wandering the halls. In real life, unless they had about 15 minutes between each class and every single club met after school every single day, there wouldn't be that many people in the halls.
Deliberate mistake: When Howard enters Figgins' password to hack into his AOL account, Figgins' inbox is obviously visible behind the login box, which simply disappears once the password has been entered. In addition, look closely at the inbox and you'll see that not only is there no mail, they are actually in GMail (the logo has simply been edited out of the top corner).