A Deadly State of Mind - S4-E6
Plot hole: Collier damages a post with his car as he leaves the driveway. His car would certainly have been damaged. The storyline is not long enough for Collier to have his car repaired. Columbo knows about the damaged post, but he never looks into how the damage occurred and never looks at Collier's car, which would have been incriminating.
Character mistake: Columbo lies on Mrs. Van Wick's bed to demonstrate that she wouldn't be able to see the toy clown if the door was closed. He has his shoes on and puts them on the bedspread and never apologises for doing so. I don't think Columbo would be so inconsiderate.
A Deadly State of Mind - S4-E6
Plot hole: Collier's relationship with Nadia Donner is wildly improper, and her husband knows about it. It seems unlikely that nothing was done about it.
Other mistake: In the cabin, Mrs. Danziger says he can pick up a pair of golf gloves in Acapulco. Golf gloves are normally sold as singles, not pairs. Very few people wear two golf gloves except in quite cold weather.
Character mistake: The crew of the ship is British. Most of them refer to Columbo as "Leftenant," but the doctor switches between "Leftenant" and "Lootenant."
Other mistake: The captain and the doctor play very fast and loose with medical records privacy, letting Columbo see them and discussing them.
Factual error: The nurse is smoking a cigarette in the sick bay. By 1975, when the show was recorded, that would not happen.
Factual error: The nurse brings Danziger a sedative and takes his blood pressure. She has her stethoscope in her ears backwards. A qualified nurse would not make that mistake. When she comes in after the murder to do it again, she has the stethoscope in correctly.
Audio problem: The clock striking wakes Columbo, and he gets out of bed. We see that he is barefoot. He takes a few steps in the barrack room, and we hear the sound he would make if he were wearing shoes. Then the sound gets louder and has an echo when we see him outdoors on the parade ground (in shoes).
Factual error: In common with a number of characters in numerous episodes, Columbo hopelessly contaminates crucial evidence - at the end of the episode he handles the surgical gloves used by the killer with bare hands.
A Deadly State of Mind - S4-E6
Other mistake: Right before Columbo tricks Doctor Collier into incriminating himself, he places Collier under arrest, even showing him the warrant. At no time does Columbo inform Collier of his Miranda Rights; meaning that Collier's self-incriminating statements would not be admissible in court. Although the series typically features Columbo cornering the culprit in a way that is unlikely to hold up in court; this particular example stands out as Columbo has torpedoed his own case. If he had not placed Collier under arrest and let him incriminate himself then the statements (in front of witnesses) would be admissible as evidence.
Continuity mistake: Columbo is eating with the commandant of a military academy. Columbo has carrots on his plate, among other things. A few shots later, the carrots have been replaced with beans. The next time the plate is seen, it has carrots on it again. Later on in the same scene, the beans are back. Finally, at the end of the scene, there is a longer shot as the commandant is leaving the table, and where the vegetables were there is now something that looks like meat. (00:36:05 - 00:42:45)
A Deadly State of Mind - S4-E6
Plot hole: (Spoiler) Mrs. Donner screams in fright just as she jumps to her death from her balcony. This seems unreasonable since she believes herself to be going for a swim (under post-hypnotic suggestion) The scream is heard by a later witness to help establish time of death. If she suddenly realized the deadliness of her act, why would she continue with it?
Plot hole: When Columbo presents the flipped photograph to Dick Van Dyke in the final scene, Columbo insists it is correct and proves DVD guilty. But Columbo already showed him the rejected fireplace photo, which shows the room arranged correctly. It also shows the victim's wedding ring, which would be on the wrong hand according to Columbo.
Suggested correction: It's all part of Columbo's bluff: Galesko wants to get rid of Columbo as fast as possible and believes himself to be too smart, thus falls for it. Galesko might have forgotten about the fireplace photo or underestimated its importance. Even if Galesko points out the rejected photo: Columbo is already bluffing about the first photo, he can very well just claim the second photo never existed or is destroyed too, leaving Galesko in the same position as he is before he incriminates himself.
A Deadly State of Mind - S4-E6
Continuity mistake: When Columbo asks if he can borrow the psychiatrist's lighter, the man has his cigarette in his mouth and a newspaper in his left hand as he reaches into his pocket for the lighter. In the next shot, as he hands the lighter to Columbo, the cigarette has jumped into his left hand with the newspaper. (00:27:40)

An Exercise in Fatality - S4-E1
Visible crew/equipment: When Janus parks his car at the health spa's service entrance, the reflection of filming equipment, including a large spotlight and the location's 9-bank array lighting, is visible on the car's surface. Then Janus lets himself in with his master key. (00:10:30)
An Exercise in Fatality - S4-E1
Visible crew/equipment: While Columbo (wearing gym clothes) is in the deceased's office, he questions Janus about the barbell and other facts in the case. When Columbo walks toward the door, the overhead moving boom mic casts a shadow on the wall at the top of the screen. (01:21:40)

An Exercise in Fatality - S4-E1
Visible crew/equipment: At Tricon, Columbo calls and leaves a message on Lewis Lacey's answering machine, then he heads to the elevator and grabs his cigar. When it cuts to Ruth Stafford at the piano, as she walks toward the door a T-mark is visible on the floor; then it's visible in following shots as well. (01:00:00)