On the tombstone of Emelia Ricoletti, it is written that she died in 1894, even though the present Sherlock says her murder took place in 1895.
Sherlock jumps down by the waterfall and his clothes wobble down during the fall. This is against physics and we should've seen clothes skidding upside due to drag created by air.
When we see the groom for the first time, he exits a building with Chinese characters to the left of the doorway. The characters used are from the simplified character set which is not possible for 1890s London as it was first introduced by the government of the People's Republic of China in the 1950s.
However, it is revealed that the entire Victorian London is in the head (mind palaces) of modern Sherlock, so anachronisms can be dismissed as him applying modern themes.
Toward the end, Holmes and Watson are in the Victorian era, circa 1885, and Watson asks Holmes to tell him more about airplanes and telephones and other unbelievable items of the future. The telephone was invented in 1876 and Holmes would have already known about them. But since only mobile phones (that of course had not been invented in the Victorian era) are used in this episode, this is probably what Watson talks about when he asks about "these telephone contraptions", and therefore it's not a goof.
When we first see Holmes he is standing next to a door frame that has been painted with Dulux Ultragloss, which wasn't available in the 1890's and, in fact, was only developed and released to the market in 1974.
However, it is revealed that the entire Victorian London is in the head (mind palaces) of modern Sherlock, so anachronisms can be dismissed as him applying modern themes.
During the carriage scene Watson has no mustache or hat in one of the cuts. However, this is clearly intentional, as by this point in the episode we have established that Sherlock is traveling back and forth between Victorian and Modern age in his Mind Palace.
When Holmes and Watson break into Sir Eustace's manor house after sighting the ghost, Holmes uses an electric flashlight. The invention of the dry cell and miniature incandescent electric lamps made the first battery-powered flashlights not possible until around 1899, more than a decade after the scene is set.
However, it is revealed that the entire Victorian London is in the head (mind palaces) of modern Sherlock, so anachronisms can be dismissed as him applying modern themes.
Near the beginning, soon after Mary appears in Holmes' flat, Watson says, "You have a case then, a new one." Holmes replies, "An old one, very old." This line and the next one are spoken far too softly for Watson to hear, since he is across the room, and Holmes is facing away from him. And yet Watson responds to Holmes' words as if he hears them.
Near the beginning of the film, "the bride" shoots her husband with a double barreled shotgun. However she is seen to rack a shell with the accompanied sound, as if she was using a pump shotgun.