Deliberate mistake: Many times throughout the show, a large bandage or makeup blot is visible on Lea Michele's right wrist, which is used to cover up her "I Believe" tattoo. This is most visible during "Don't Rain On My Parade" and her "Poker Face" and "For Good" duets.
Deliberate mistake: When Ryan and Esposito give Beckett the surveillance tape of Sophie and Uncle Teddy, Beckett's computer reads the disc and plays the tape incredibly fast. Faster than any desktop PC I've ever seen.
Deliberate mistake: Because the opening narration was adapted from the original American series, the phrase describing the role of the Crown Prosecutors (e.g. "prosecutors who prosecute") is redundant.
Deliberate mistake: The same outside shot of the Big Nugget Riverboat Casino is used over and over just zoomed in or zoomed out. Notice how the exact same cars are in the exact same place each time. (00:08:05 - 00:10:25)
Intermediate Documentary Filmmaking - S2-E16
Deliberate mistake: This episode is meant to be entirely a documentary made by Abed. At the end of the episode, there's a shot where the gang is looking into Pierce's window, except the camera is right in front of the window they're looking through. If Abed was filming this he would literally be right in front of the gang, completely obscuring their view through it.