Factual error: An important plot point is that diamonds do not burn and can be recovered from the ashes of a cremated body. This is completely wrong - diamonds would combust freely at cremation temperatures, and since they are pure carbon they don't even leave any residue.
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?
Factual error: In the middle of summer in southern California, water spread thinly on an outdoor ceramic surface would start evaporating immediately and would be gone in about fifteen minutes. The surrounds of the pool would be bone dry by the time Columbo arrived at the scene, nearly an hour after the murderer sprayed the water there.
Continuity mistake: When Buckner's car leaves the chemical plant, it's a stretch limo with three bench seats. When it explodes on the mountain road a few scenes later, it's only a two-seat limo. (Buckner, who was initially in the distant rear seat, is suddenly close enough to reach up and take the cigar box the chauffeur hands him from the front seat. The entire middle section of the limo has disappeared.)