Factual error: At the onset of the baseball game, take a look at the Angel's catcher as Mariner Jay Johnstone steps up to bat. You will see he is wearing just a standard baseball hat under his catcher's helmet. MLB rules requires some form of certified (protective inside and out) plastic helmet for catchers, and thus this would not be allowed.
Continuity mistake: When Frank is fishing the pen out of the fish tank and the fish gets attached to his nose, there is visible blood. When the camera cuts back to Frank, the blood is gone.
Visible crew/equipment: At the end of the film when the guy in the wheelchair is rolling down the stairs you can see a burst of gas (presumably from a CO2 cylinder) fire the man up.
Visible crew/equipment: In the opening sequence, the camera's shadow on which the car roof and siren are mounted is visible as it enters the house.
Continuity mistake: As Frank and Jane head back to her apartment, a woman passes them by with her dog. When the camera cuts to Frank several times in the scene, you'll notice that the woman and the dog are in several different places and end up turning the corner twice.
Continuity mistake: When the "I LUV YOU" picture is viewed in Nordberg's hospital room, the photo has a vertical/portrait orientation. When Frank shows the picture to Jane, the photo has a horizontal/landscape orientation.
Other mistake: Pay attention to one of the early innings of the baseball game as one of the Angels players hits a home run. Though probaby intentionally set up this way as to avoid any problems of "logo patents" and such with the real life MLB California Angels team, he is wearing a helmet without the team logo on it.
Continuity mistake: When cutting into the cake with the queen's portrait on it, the waiter slices across the inner corner of the queen's right eye. However, the two pieces of cake served to Jane and Frank both have complete and uncut eyes.
Audio problem: Right at the scene where Reggie Jackson points the gun at the Queen, Drebin fires a shot off at him from his watch. It goes astray, hitting some fat lady, and we hear him saying "whoops!" but his lips don't move.
Revealing mistake: During the car chase, the doc's (already busted) car collides with a missile transporter. The explosion occurs just before the car impacts with the vehicle, and looks more like the burst of a flame thrower than a real explosion.
Continuity mistake: While Frank Drebbin is saying the words "dew glistening on the grass" you can clearly see a woman in the audience at the bottom of the screen repeating every word he says.
Continuity mistake: Just before Frank accidentally nudges one of the fishermen into the river with his car, if you look in the background there is a police car. In the next shot of Frank getting out of his car on the pier, the police car is now noticeably much closer than in the previous shot.
Continuity mistake: In the scene where Frank is in the driving school car chasing the doctor, right after Frank says, "I know a shortcut. Turn left here", we see two brief shots of the doctor's car, one from the inside of the car and the other from the outside as he is turning around a corner. In both shots the rear window is shattered. However, it is only four shots later that Frank shoots at the doctor's car, shattering the then intact rear window.