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)
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.
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.
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)
Other mistake: When Rosanna's body is brought to the ship's medical room, the doctor confirms that there are no bruises on the body, yet she had been forcibly punched in the face earlier that day by Mr Danzinger (Robert Vaughn). The punch was hard enough to knock her over therefore a bruise should be visible, even through make-up.
Plot hole: The cruise ship is sailing from LA to Mazatlan in Mexico. Looking at a map, the ship will hug the west coast on its journey therefore is never far away from land. The murder takes place on the first evening of the cruise only a few hours after setting sail by which time they won't be far from San Diego. Why didn't the captain order the ship into San Diego or another suitable harbour and let the police take over the investigation?