Continuity mistake: After receiving the third pip. Dr. Watson turns on the television to the news. During the news segment, it states the age of Connie Price as 48. Then in the scene immediately following, DI Lestrade says she was 54.
Continuity mistake: In the opening scene when John is going to the refrigerator, a white skull can be seen on the left side of the kitchen table amongst all Sherlock's science equipment. When he closes the refrigerator door again after discovering the severed head, the skull has disappeared. As he's leaving the kitchen, the skull reappears. (00:03:25)
Character mistake: In the opening scene, Sherlock makes a point to correct Barry's poor grammar. But later, when talking to Watson, Sherlock uses poor grammar by saying "who's sleeping with who." It should be "who's sleeping with whom." Not a mistake Sherlock would make.
Factual error: In a couple of scenes with the mobile phone when Sherlock is talking to the girl and she hangs up, the soundtrack has a audible "dial tone" inserted. Mobile phones don't have dial tones.
Character mistake: Lestrade mentions that there could be some "poor sod covered with semtex on the streets somewhere" when talking about Moriarty's victims, but semtex does not go off via gunshot, much like C4. But that is what happens when one of the victims dies however, so the explosives are not semtex.
Suggested correction: The term "Semtex" is often used to refer to any plastic explosives, much like the term "Hoover" for a vacuum cleaner or "Kleenex" for a tissue.
But that doesn't negate the mistake.
Continuity mistake: When Sherlock pulls the gun on Moriarty in the pool scene, he is holding it with two hands but then puts one hand down. In the next shot, he has two hands on the gun again. Then again in the next shot he only has one hand on the gun.
Character mistake: When Sherlock receives the phone, it is in a hand-made paper envelope. He says "Nice stationery. Bohemian" and specifies "From the Czech Republic." The paper is Losin, which is produced in Velke Losiny paper mill, the oldest in Europe and the only Czech one. However, it is not located in Bohemia, but in Moravia, which is a different region. It is an equivalent of calling Glenfiddich "English whisky from Britain." An error Sherlock would not make. (00:11:45)
Continuity mistake: At the end of the episode when Sherlock takes the explosive jacket off John, he throws it away from them both. You can see the jacket change places during each camera shot. (01:27:10 - 01:28:02)
Continuity mistake: When Sherlock pushes the door open to leave the museum, it closes from the other side when reentering the frame. (01:00:56)
Continuity mistake: When Sherlock goes to look out the window at John leaving (after the tiff about the head in the fridge), his dressing gown is hanging off his right shoulder, exposing his t-shirt. Next there is a wider shot of him from behind, and his dressing gown is on straight, covering both shoulders properly. Then there's another closeup and it's off the right shoulder again. (00:05:40 - 00:06:10)