The officer seen in the police car that drops Mac and Dee off in the "bad neighborhood" is not either of the two officers Mac and Dee are assigned to in the police station.
Right after the theme song, when the gang is talking around the pool table, the position of the various pool balls changes from shot to shot.
When the gang is first seen driving in the cop car, they state that they're stopped at a red light at the crosswalk, yet a car to their right is seen driving right past them.
When Dee is using the bat on the bum it's obvious the bat is made of rubber since it bends around the bum's arm.
When Charlie walks into the police department to hand in the cassette tape, he bumps into a New York State flag.