Dr. Watson is repeatedly described as having "no ear for music". Yet in two previous films of the series, he is seen both singing and playing the tuba.
The auctioneer read the address from the log to Colonel Cavanaugh (the villain), not Mr Holmes. When Holmes and Watson visited the auctioneer, the dialogue had Holmes mentioning the (incorrect) address.
In the close-up of the woman's hand holding Sherlock Holmes's calling card, the fingers have red nail polish. In the very next shot, the woman does not have nail polish.
When Sherlock goes to see the auctioneer to ascertain who purchased the music boxes, he reads the purchasers address from the log and says it's on "Hampton Way" In the next scene he visits the house on Hampton Road. He also mentions the "correct" street, Hampton Road in a subsequent scene.
When Holmes finds the Kilgour child tied up in the closet, she's gagged by a large scarf that covers her nose and hangs down her neck. Seconds later, when Holmes lifts the girl out, the gag is below her nose and ends right above her chin. Clearly, it is two different scarfs that have been replaced in between shots.
The bus has a left hand drive, American style. All the cars and taxis have right hand drive, which is correct.
The premise of the film is that the only way the prisoner in Dartmoor can communicate with his colleagues outside the prison is through the music boxes. So how does he get the information to them about when and where the boxes are to be sold? This can be explained if the prisoner can pass message outside through an intermediary, but for safety, he doesn't want to directly inform the intermediary about the location of the plates.
Dr. Watson refers to having met the King of Bohemia, which does not make historical sense, as the monarchy in Bohemia (land of the Czechs) was abolished in 1620. This is an allusion to the source story "A Scandal in Bohemia" which gave that land a tongue-in-cheek fictional history and politics.
The reflected image in Hilda Courtney's makeup-mirror is the same orientation as the real scene, rather than being reversed, as it would be when viewed in a mirror.
It seems unlikely that Holmes would be playing "Danny Boy" on his violin, a sentimental ballad more associated with the Irish community and popular with American film audiences at the time. The tune is not mentioned in any of the stories, where he played classical music, more likely Mendelssohn.
When Mrs Courtney leaves 221B Baker Street after her visit to Dr Watson, we can see that the paintwork on the inside of the front door seems to be extraordinarily shabby and scratched.
When Holmes is playing the violin in his quarters, his hands rarely move on the neck of the instrument. He is not really playing it.
The first scenes show prisoners wearing clothes with wide arrows, a design introduced in the 1870s but discontinued in 1922.
One of the scenes depicts a "General" omnibus carrying passengers on a normal route. The movie is set post-WWII London (1946). The London General Omnibus Company went out of service in 1933.
Watson falters, an instant, as he opens the lid of the musical box after removing it from the biscuit jar, causing the lid to mostly fall closed again for a second, before he lifts if open fully. Yet, the tinkly music continues to play seamlessly, without any interruption from having the lid closed again momentarily.
A crew member can be heard laughing as the Auctioneer describes the music box as sounding like Bow Bells.
When Holmes is being driven to the location of his demise, he requests a cigarette from Cavanaugh, who takes out a cigarette case from a pocket inside his coat. At that moment the vehicle lurches and Holmes simply puts his hand inside Cavanaugh's coat for a second and miraculously extracts the key to the handcuffs.