When Keeler is connecting the clamps to the communications line to ping it to find the location of Cohen's central book, he first clamps a red wire to the line. In the next shot, he is getting ready to clamp the black wire near the red wire on the same communication line, but when the camera changes angle and he finishes connecting the black wire, the red wire that was already connected is no longer there.
The kidnapped girl is thrown on the bed and as John O'Mara comes into the room, the bed is put up in order to hide her, meaning that she should be upside down. When she is yelling for help, she is standing up.
The film gives the impression that the Gangster Squad were responsible for Mickey Cohen's arrest. In reality, Cohen was arrested in 1950 for the more mundane crime of tax evasion. He was also not sent to Alcatraz until 1961, over a decade after the film's time frame.
The movie suggests that Mickey Cohen had Jack Dragna murdered in his own house, but in real life Jack Dragna died of an heart attack years later. The movie is set in the late 40's/early 50's while Dragna died of the heart attack in '56.
When O'Mara and Wooters are sitting on the porch, the bullet holes that surround the window frame have charred edges. That would have happened if the shots were fired point blank, but in the event they were fired from a substantial distance away, so there should have been no charring of the wood.
When the two civilian men are playing Chinese Chess in Chinatown, the pieces clearly do not represent an actual game because some of the pieces are in positions that are not possible in a regular game.