When Capt. Breen is interviewing Paul, he gets up from his desk and sits down closer to Paul. A shadow of the boom microphone is visible behind Paul, moving onto and then off the door frame behind him.
When Martin flees from his bungalow into the sewer system, the first shot shows him running in the sewer with a torch and a bag in his hand. This is the same shot as used earlier on in the film after he started robbing liquor stores. (In this later scene, he did not have a bag when he fled the bungalow nor when he entered the sewers.)
After examining the bullet casings, Lee tells Capt. Breen that the same gun was used to shoot Officer Rawlins and Detective Jones. But Roy actually used a revolver to kill Rawlins and a .45 automatic to shoot Jones.
A revolver ejects its spent casings only if the shooter does so manually, which Richard Baseheart didn't do, therefore the police could not have even had them to compare with the fired casings from the automatic. An even bigger blunder was the photo comparing the two fired shells, they were clearly marked .380 auto, an entirely different round than the .38 special or .45 auto that the police claimed were fired from the same gun. So now there were three different guns used to create this technical error.
In addition, when Jack Webb shows his blow-up photos, all three ejector markings are in the precise same location on the cartridge head, a statistical impossibility. In fact, all three are blow-ups of one photograph.
Despite wearing both shoulder holsters and belt ones, the police in the film never unbutton their coats to remove their handguns. Since drawing your weapon is more important than remaining fashionable, it's a certainty that they would unbutton before drawing.
The goof item below may give away important plot points.
There is additional lighting from some source(s) when the police are chasing the antagonist through the sewers. This is especially noticeable after the antagonist loses his flashlight. Obviously, there would be no illumination at all in a 1940s stormwater/sewage unless people brought it with them.