The tyres on Superintendent Durk's car in the London scenes are of a different make than those on it in the scenes "in the country" (that were actually shot in the US).
In the first scene, Nicholas stops his car in front of the theatre. There are road markings in the middle of the street in one shot, but in the next shot, they are gone. There's also a completely different set of people on the pavement/sidewalk.
As Columbo sits reading the Evening Echo on a bench, an elderly man is feeding the birds next to him. His position changes noticeably over camera shot changes as O'Keefe walks over and attracts the Lieutenant's attention, from upright to leant over.
During Nicholas Frame and Joe Fenwick's tuppence-a-point darts game in The Kings Arms, a man stands at the back watching the match. His stance alters notably between shots after Joe takes his second throw.
Sir Roger Haversham's house is supposed to be in England, but everyone drives their car on the right-hand side of the drive.
Chief Superintendent Durk's car has an incorrect number plate format; it has 5 numbers, whereas the maximum number of digits for a GB registered car has never been more than 4.
The coat of arms behind the coffin has a coronet (apparently of a European Count) rather than a helmet, which Sir Roger would use as a knight.
Det. Durk refers to Sir Roger Haversham as his wife's uncle, twice removed. This relationship applies to cousins, not uncles. Technically, he should have referred to as her great-great uncle.
Near the end, when two police officers are putting Frame and Stanhope into a police car, they are wearing all black caps. Most police in the UK wear hats with a black-and-white checkered pattern (Sillitoe tartan); there are some variations, but never all-black caps.
(at around 1h 15 mins) Det. Chief Supt. William Durk does a little skip in his step. However, this should not be considered a Goof as many people have little quirks that may include occasionally skipping, talking to themselves, hopping, etc.
When Columbo starts taking photos outside Buckingham Palace, a person in the crowd is dressed in the uniform of a British Rail ticket collector/platform staff with a dark round cap, black waistcoat, and maroon sleeves. However, this is not a Goof as there is no reason such a person should not be in the crowd.
The basement of the wax museum is Paul Rifkin's nightclub in Étude in Black (1972).
This, however, does not constitute a Goof. Goofs need to be contained within the movie/show being viewed; consistency across episodes or sequels is not required, as viewers may not have seen earlier shows.
After Frame and Stanhope (Richard Basehart and Honor Blackman) tie a bicycle to the back of their car and drive off, all of the cars parked on the other side of the street are facing the way they would be parked in the USA (i.e., as if they were driving on the right).
When Frame and Stanhope are strapping a bicycle to the back of their car, the same extras can be seen walking behind them, first from right to left around the corner, then almost immediately return back around the corner from left to right. It appears no attempt was made to to blend in a little better (e.g., lose carried bags/coats or umbrellas).
Columbo talks to the actors about the murder in their dressing room after a performance. He quotes from a folded newspaper what the weather was, and the paper has a banner headline that includes the words "King" and "Constitution", implying this would be concerning Edward VIII's abdication in 1936. The Masthead includes the word "Express", but it is not of the London Daily Express, indicating this likely was a prop.
The pearl dropped in the museum lands on carpet, but it sounds as if it was bouncing on a hard floor.
When Frame and Stanhope are pushing the trunk out of her dressing room, shadows are seen on the floor of either the moving camera or crew.
During a montage of Colombo and Chief Supt. Durk on a boat traveling under Tower Bridge, the camera pans from Peter Falk to reveal a prominent white microphone.
(at around 1h 20 mins) Durk tells Colombo that Sherlock Holmes was, perhaps, Scotland Yard's most famous detective. This is false for two reasons: first, Holmes was fictional, which Durk, of course, would know; second, in Sir Arthur Conan Doyle's mysteries, Holmes was never in Scotland Yard; he operated independently.
Lilian Stanhope addresses Columbo inconsistently, pronouncing his rank as 'Loo- tenant' (the American pronunciation) and 'Left-tenant' (the British way). An English actress of her standing always would use the British pronunciation.
The sign outside the theatre advertises "In memoriam of Sir Roger Haversham". In Latin, "In memoriam" means "in memory of", so the sign translated into English would say "In memory of of (sic) Sir Roger Haversham".