The single bullet inside Miles' gun is not positioned correctly to fire on the first shot (which it is seen to do).
When Miles is hanging in mid-air in the opening chase, he pulls out a circular saw to cut through the rope he is hanging from. He cuts the rope in front of him which would have made him fall backwards, but he falls forwards instead.
The remote controlled car and glasses that Miles has been using both disappear after he enters the video room in the police station and the lights are turned on.
When Miles first enters the department as a detective, Detective Larson is shown loosening the handcuffs on a suspect. The shot shows that only one handcuff is released, but yet when the suspect runs, there are no handcuffs on him at all, when there should be both hanging from one wrist.
During the chase scene, when Miles does the large jump on the bridge, when the camera cuts to the inside of the car, to see Carlson screaming, look out the windows and you will see that the car is on the road, then when the camera cuts back to the outside of the car, it is airborne again, then it lands.
When the cable was cut, the pulley, along with Miles, should have dropped to the ground. There was no momentum for a swing in either direction.
When Miles is talking to his girlfriend after coming out of jail, there is a person on a bike visible in the background. The person is in a riding position, but the bike is not moving. If you look closely, you can see that person is leaning on his left leg.
When Miles launches the police car into the air, shots from inside the car as it flies show the guardrail of the bridge outside, ruining the illusion that they're in a car flying through the air.
Miles, Tully and Deek are in the back of the truck with the heroin. When the big magnet picks up the truck, Deek is still able to hold the metal gun in his hand.
When Miles is dressed as a pizza delivery man, he opens the front door of the police station and a wall holding a bulletin board is visible just in the door. Though when showing a shot of the interior after he walks in, the room is an open space with no dividing walls in front of the doors.
The camera can be seen reflected in the glass door as the guard is shining his flashlight through it at the beginning.
Before Miles enters the police station dressed as a pizza man, the reflection of a crew member dressed in blue jeans is visible in the front door.
After Miles leaves the meeting and enters the office, he reaches for a donut in a Dunkin' Donuts box when in fact, during the filming of the movie, there was not a Dunkin' Donuts in the LA county.
After Miles leaves the police station dressed as the pizza delivery guy, the reflection of the boom mic can be seen in the window behind him as he dances.
When Miles first enters police station there is a red hair in ponytail man with a beard in cuffs entering the scene with arresting officer holding him.
This exact same shot, with same actor wearing same outfit is seen later in the film with same arresting officer.
It's not used as a gag- it's laziness, as they just used same shot twice.