The sign outside Dr. (John) Watson's office reads G. WATSON, M.D. Either nobody noticed the art director's mistake or, more likely, they didn't want to go to the time and expense of making a new sign.
In the Stepney Gas Chamber, Moriarty's men are warned not to smoke because it might start an explosion. Yet later, in the same location, Sherlock Holmes and James Larrabee are sitting at a table, across from each other, and not only are they both smoking, a lit lantern is on the table between them.