When Dick Van Dyke picks up the gun to shoot Antoinette Bower she reacts far too quickly to the "sight" of the weapon. She could not possibly see the pistol, as it is still shielded by the lid of the briefcase that it was stored in.
The fastidious driving examiner Mr. Weekly tells Columbo about a driving exam accident he had had "yesterday". This contradicts the DMV clerk's earlier statement that Mr. Weekly had come back to work that morning after having been out of town on previous days.
When Columbo is explaining to his Captain that it would be unusual for somebody to use cabs rather higher a car he begins to ad-lib a little too much and the stern faced Captain struggles to contain his laughter.
When Sgt. Hoffman hands the ransom note and photo to Columbo, he tells him to be careful as the labs boys haven't examined it yet. At which point Columbo handles them with his bare hands and even pushing his finger on the photo to point something out.
Sgt Hoffman gets a phone call and summons Columbo back into the office. When Columbo comes through the doorway, his tie is outside his collar. In the next shot, however, the tie is completely inside the collar.
The glue cap/brush that Columbo is using in the evidence room at the end of the show changes positions on the desk between shots. First it is closer to Columbo towards the edge of the desk. The shot changes to Galesko, then back to Columbo and the cap is now farther away, closer to the bottle. Then at the end of the scene, the cap is back on the bottle even though no one screwed it back on.
Columbo sets up a trap for Galesko by making a reverted copy of the photo he found in the fireplace. Since he learned before, that every negative can be printed reverse (wow!), he uses this to shatter Galesko's alibi. First, every child knows, that instant cameras don't produce negatives! It's no surprise that Columbo doesn't know this, but Galesko as a professional photographer should know. So what does he think he's finding inside the camera? There's nothing but unexposed papers inside. The second point is, that taking the camera doesn't prove that he knew it before. Since Columbo showed the photo from the fireplace to Galesko before, he as a photographer would know that is was taken with an old Polaroid Land Camera (b/w paper available only), and there is only one of this kind on the shelf. So it's naturally that he takes the right one, although the purpose is complete nonsense.
As Dick Van Dyke brings the handgun from behind the briefcase in his left hand, it is a German Luger, made mostly between the early 1900s through WWII. This type of gun has a toggle action, which is held open and up after the last shot or when one cocks the gun on an empty magazine, and it's up just before he shoots it. The gun cannot be shot in this condition unless the toggle is pulled back and released with a magazine that's got at least one round in it. This is akin to trying to shoot a modern gun while the slide is locked back after the last shot. It must be released first on a magazine with rounds in it. If Locked up and open the gun won't even go "click".