Plot hole: The alien-possessed Mason steals Captain Lauritzen's fingerprints by pressing his fingertips to the captain's and "heat transferring" the prints. Big problems here: those prints would now be backwards and for the wrong hand. But when the computer later reads the prints, it fails to notice this and passes him through security as "Identity Correct." (00:23:30)
The End AKA Encased in Concrete - S5-E13
Plot hole: The Goodies office is encased with concrete. The Goodies survive for over 75 years. How did they get enough air to survive all that time?
The Messiah on Mott Street - S2-E38
Plot hole: No one in Goldman's house finds it at all strange that a mailman arrives at the door with a letter after midnight on Christmas Eve. Nor, a short time later, does the doctor or anyone else passing by at this very odd hour wonder why the same mailman is collecting mail from a street box at dawn on Christmas Day.