Holmes grabs a lamp off of the wall and puts it down by the hole in the floor. A short while later the lamp is back on the wall, but a short while after that it's back on the floor in time for Watson to trip over it.
When Watson throws the darts into the magazine on the door he is shown to throw four darts, but when he grabs the darts and the magazine off the door he only removes three darts.
When Inspector Lestrade leaves the Royal Mint in a carriage, there is just him and the driver. When he arrives at Giles' house, he now has two policemen with him and the driver.
When Lord Smithwick and Inspector Lestrade are leaving Holmes' apartment, Holmes has partly closed the door with the Inspector partly turned towards Holmes. The next scene has the door fully opened with the Inspector facing Holmes. Then Holmes closes the door in the Inspector's face.
When Holmes is writing on the chalk board, the writing of 'Moriarty' changes between shots (most notable is the 'M')
The scene between Watson and The Strand publisher shows the label "Norman Greenhough - Publisher" on the office door. The Strand was founded by Sir George Newnes in 1891, and he was also Publisher until 1910. The first Editor, from 1891 to 1930, was Herbert Greenhough Smith.
The real Leslie states she was in the revue "Les femmes faux". "Faux" is the masculine form of the adjective and is in the wrong position. The correct form should be: "Les fausses femmes". (Unless, of course the writer really knew French and was making a joke.)
The ticket master who sells Watson the tickets for the trip to London from Lake Windermere pronounces the word 'class' in 'first class' the American way 'classs'. A Brit would pronounce it (phonetically) as 'first clarse'. However this is an over-simplification. The pronunciation of class depends on the accent of the Brit: any Brit born north of approximately Northampton would pronounce class as an American does; southern English speakers would say "clarse".
During Holmes and Moriarty's sword-fight, Michael Caine's fencing double can clearly be seen from time to time.
When Holmes and the would-be kidnappers are fighting on the stairs several of the balusters can be seen to be swinging free and are not fixed at the bottom.
The mail at Peter Giles' place has fallen to the centre of the entrance door as opposed to the side directly under the mail slot.
The postcard from Lesley has a picture of the Eiffel Tower occupying one entire side. Postcards of this type did not exist anywhere until 1901 or 1902, and in France not until 1904.
Holmes mentions the Loch Ness Monster. While modern enthusiasts have pored through history to find 'evidence', the modern craze and the phrase "Loch Ness Monster" began in the 1930s, well after this film, set in the 1880s.
They mention the mambo as "a festive Caribbean dance" when in fact the mambo wasn't invented until well into the 20th century (probably as late as the 1930s).
Moriarty uses arming lighter when on the docks. Ferrocerium (which is how flint lighters work) was not patented until 1903.
Around 1 hour into the film, when Lestrade and Smithwick leave the room (at 221B) towards the camera, the shadow can be seen on them, especially on Lestrade's back, which could be filmmaking equipment. But the shape is a curve with a hanging decoration, showing that the shadow just comes from one of the lighting fixtures on the wall.
One comedic scene has a character introduce herself as Leslie Giles. She faints at the news (boorishly broken) that her father is presumed dead, and she gets carried to safety by Doctor Watson. She's later exposed as an imposter who knows almost nothing about the father. It's never addressed why she would faint, or how she might have faked it well enough to fool a brilliant medical man.