When Eric is confronting Principal Anderson and says this is an issue from 1983, as he flips through the pages throughout the scene, you can clearly see a picture of Bret 'The Hitman' Heart in the bottom right hand corner in the magazine. Bret Hart came into mainstream wrestling much later than 1983.
When Billy is dialing his classmate Danny McGrath, the sounds of the buttons beeping does not match up with his fingers. Also, we hear the sounds of the phone ringing long before Billy is even finished dialing.
In the bus on the way to Billy's third grade field trip, there are three children between Billy's and Ms. Vaughn's seats when we see them from the back of the bus. When Billy is talking with Ms. Vaughn and we are looking toward the back of the bus, there are only two children between Billy's seat and Ms. Vaughn's.
When Billy and his friends throw the pickles on the window for the race, the one on the far right (Billy's) has the lead. In the shot after Billy proclaims himself the winner, the pickle in the middle has the lead.
When Billy and Eric face off on a math problem in the academic decathlon, Billy writes "Eric drinks his own pee" on his portion of the blackboard. Traces of erased portions of this phrase from previous takes can still be seen on the blackboard.
When the kids come to Max's house, a shadow can be seen passing behind him in the hallway. This is from a deleted scene when one of the kids dresses up in Max's old wrestling costume and runs out on the lawn.
In the quiz show scene, Billy recites the "To Be or Not To Be" soliloquy from Hamlet with a skull in hand. Hamlet doesn't recite that soliloquy with a skull in hand. He only speaks with a skull in hand in the grave-digger scene at the end of the play. However, the skull was probably just a prop as part of Hamlet's costume because that is how people know Hamlet.
When Principal Anderson is talking to the kids at his doorstep, its cuts to the kids outside sitting on his lawn and then back to Principal Anderson. If you look in his house, you can see the shadow of one of the cast members running to the left.
When the O'Doyle family station wagon skids off of the road toward the edge of the cliff, the driver is deliberately turning the steering wheel in the direction of the cliff, and a bracket and cable are visible on the front end of the car.