The scenes shot on the roof of St. Barts are physically different from the street view of the same area. In the rooftop scenes, the front ledge of the building is about twenty feet wide. From the street the ledge is around 80 feet wide.
When Moriarty is getting ready to steal the Crown Jewels, the position of the gum he places on the glass to hold the diamond changes between takes. Sometimes it is by the bottom of the "E," sometimes it is to the right of the "T."
When Sherlock and Jim are at the roof top and Sherlock laughs and comes back to Jim to tell him what he missed, it's clearly visible that his scarf is a little messed up, but when the shot is focused on him 2 seconds later, the scarf is perfectly straight.
In the scene when Sherlock is analyzing the bottom of the kidnapper's shoe, he finds traces of Glycerol/Glycerine/Propan-1,2,3-triol. The Glycerol molecule contains 3 hydroxyl groups each bonded to three carbon atoms. The show misrepresents the molecule as having 2 hydroxyl groups, and one unknown OE group, which is in fact not even a chemical compound.
In the courthouse scene, where Moriarty asks security to reach into his pocket, she does so and places a piece of gum into his mouth. A trained court security officer would not do this, as this would place them in a position of disadvantage. In addition, as Moriarty was in custody, his clothing would be searched and contraband such as gum would have been disposed of. Gum especially is a security concern as it can foul locks,etc.
When running from Scotland Yard, handcuffed, they are almost run over by a route 74 bus. Although route 74 terminates at Baker street, it doesn't display Baker Street Stn as destination but simply Baker Street.
Episode opens with Watson narrating the story to his psychiatrist, mentioning that his best friend Sherlock Holmes is dead. Story goes back 3 months to the day of Richenbach Case, followed by Jim Moriarty's crime spree, after which he get arrested. It is mentioned that his trial took 6 weeks, and story moves forward 2 months after the trial to continue Watson's adventures with Sherlock. However, by this time Sherlock should already be dead, and Watson's psychiatrist appointment must already been in past.
Watson's first view of Holmes after he has jumped is partially obscured by a truck. In the shot where he is shown being knocked over by a cyclist, the truck is no longer in shot, and the bus stop further down the street is clearly visible, however while he is falling, the camera position shifts and then goes back to the view along the street in which the truck is back, and partially blocking the bus stop.