At the end of of the first Naked Gun movie Frank is promoted to Captain by the mayor, however in this film that promotion is disregarded for the continuation of the in-joke involving Frank being both a Lieutenant and a Sergeant.
When 'George Bush' is trying to persuade Frank Drebin to start a new police squad, his glasses sit tight on his right ear, then jump about two inches above the ear, and then return to sitting tight on the ear.
When Frank and Jane are trying to crack the code to stop the bomb going off, Frank's jacket is eaten by the machinery. In the following scene when he is up on the stage making a speech, he is wearing his jacket.
Towards the end of the film, Jane is taken hostage. When she first appears as a hostage, she is gagged so that both of her lips are still visible. In subsequent scenes, the gag covers the whole of her mouth.
In the scene where Nordberg is trapped under the car, you see a few shots of what he is about to hit. We see him about to hit a traffic cone and tumbleweed but instead he hits a cactus with the traffic cone and tumbleweed further ahead in the road.
Whilst Norbert is being dragged under the front of Frank's car and is hitting objects left in the road, the car drives past the camera and you can clearly see that Norbert has two animatronic legs attached to the front of the vehicle.
Anabell Brumford is introduced as Washington D.C.'s police commissioner, with the rank of Captain. The Metropolitan Police Department of the District of Columbia (Washington D.C.'s police force) doesn't have a commissioner and is headed by a Chief of Police. Captain is usually in the middle of the rank structure of American police departments, including Washington D.C.'s.
The picture of the Titanic hanging among other "big failures" at the Loser Bar is actually the Lusitania, a British liner sunk by a German U-boat during WWI: note the propellers (four instead of three), the different shape of the stern and the fact that, in the picture, it is daytime; although most of these aspects can only be told in a frozen frame.
The police forensics expert says that his team have discovered a dinosaur footprint outside the institute that he describes as a major find from the Paleolithic Era. The Paleolithic Era is the Old Stone Age, which took place about sixty million years after the extinction of the dinosaurs.
In the opening scene, Captain Brumford mentions that Frank's rank is Lieutenant. When Frank begins his opening narration, he mentions his rank as Sergeant. This is not an error but a running joke in the series that comes from the original Police Squad! TV show, where Drebin would be referred to as several different ranks through an episode of even in the same sentence. Frank's full, official title is "Sergeant Frank Drebin, Detective Lieutenant Police Squad", which is a fictional amalgamation of three different ranks of police officer.
During the shootout on the roof, the number of shots exchanged by Frank and the gunman far exceeds the capacity of either of their pistols. This is a gag in all Police Squad episodes and Naked Gun movies - an exaggerated capacity to the weapons used.
Right after Nordberg blows up the door to the roof access at the hotel, Frank enters and leaps over the railing behind a computer panel. When the stuntman lands, his shadow stays in the same place while Leslie Nielsen comes out.
When Ed is looking through Jane's sketch book, he acts shocked at how the sketch artist drew Jane in exotic outfits, then he flips the pages. The drawings are facing the camera, and there is obviously nothing drawn on the page facing Ed.
Right after Frank, Nordberg, and Ed reach the roof, they are fired on by 2 gunmen, one with a submachine gun and the other with a pistol. The gunman with the pistol fires 4 rounds at the three detectives, but only one shot produces a muzzle flash.
In the scene in Jane's apartment when she climbs into the refrigerator you can see that her high heels are a normal length. Later, as she tells Frank to let himself out, her shoes' heels have grown to a ridiculous size.
On the roof Nordberg fires his huge gun, firing an artillery shell through the wall of the building. However, when the explosion is shown there is no sign of an artillery shell and the explosion clearly originates from inside the building.