Continuity mistake: When Kay retrieves the gun from the elevator ceiling, they switch shots several time looking up at the ceiling and down from the ceiling. In one shot looking up, the barrel and grip of the gun are over the edge of the ceiling tile. In the next down shot, just the barrel is near the edge and the grip is inches away. Back to an up shot and the gun returns to barrel and grip over the edge.
Factual error: Devlin proclaims himself to be a staunch Irish Republican but he refers to the Northern Irish city of Derry as "Londonderry." This is the name imposed upon the place by the British and is despised and rejected by Republicans. Devlin would refer to the city as "Derry."
Plot hole: Columbo is in the Pauley's hotel room trying to figure out how the bottle landed upright next to his body after he was shot. Devlin - the murderer - turns up unannounced and uninvited. How did Devlin explain that he knew where the victim was staying when he was murdered? Media reports might have mentioned the name of the hotel but there is no way they would have included the room number.
Plot hole: Just before Joe Devlin kills Vincent Pauly in the hotel room, he picks up the Full's Irish Dew whisky bottle with his bare hands. The same bottle falls to the floor when Pauly is shot, and Devlin then kicks it over to the body. Devlin didn't wipe the bottle clean at any point, therefore it will be covered in his fingerprints, but this is never mentioned throughout the episode. There's no way the police would overlook such a basic clue.