Short Fuse - S1-E7
Continuity mistake: A chauffeur is driving Mr. Buckner to a mountain cabin. It is 7:30 pm and a very heavy rain is falling. The car explodes, killing both men. When their bodies are located the following morning, the road and the surrounding hills are completely dry. There are not even any puddles.
Continuity mistake: During his lunch with Brimmer, Columbo's wine and water glasses switch places several times. (00:00:50)
Character mistake: This episode opens with James Ferris typing the final chapter of a mystery novel. He is using all-capital letters, No professional writer would do that.
Suggested correction: It's not meant to be all capital letters, but block letters. Just look at the title card and credits in the opening - they're block letters too.
Short Fuse - S1-E7
Plot hole: We are told several times that the road to the mountain house is very narrow and difficult to negotiate. How did the very large crane get there so quickly to lift the bodies after the crash?
Character mistake: Columbo talks to the cardiologist. The doctor gives him information about his patient, Mr. Williamson. Columbo has no warrant, so the doctor is violating privilege. He wouldn't give that information to Columbo.
Dead Weight - S1-E4
Other mistake: Columbo goes into a diner and has a bowl of chili. The cook asks if he wants his coffee black. Later he asks if he wants cream and sugar in his coffee.
Continuity mistake: When Columbo meets Goldie for the first time she is having a massage. She then proceeds to dress by tying a dressing gown with a bow at the front. In the next shot the dressing gown is open with the tie at the back.
Continuity mistake: Colombo is talking to Mr. Kingston in the building entry after the reading of the will. A girl with red and white striped sleeves walks out of the building. A minute later she walks out of the building again. (01:15:50)
Continuity mistake: When Columbo confronts Markham in the lecture hall and helps him erase the chalk board, most of the word "architecture" remains, visible over Columbo's shoulder. In the next shot, it disappears. (00:31:00)