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 among 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.
When Sherlock exits the museum, he pushes the exit door out, but when it slams shut moments later, it closes from the hallway to the door frame. Exit doors are required to open outward, but the door closing in the second shot would have opened inward, against safety regulations.
After it is flung to the ground, John's jacket containing the explosives moves backwards and forwards on the poolside between shots.
After Sherlock stops shooting at the wall, he swipes his hand upwards where he shot, pushing the now torn wallpaper, up. In the next shot we see of Sherlock, the wallpaper is intact.
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.
It's a common movie myth but firearms cannot be used to set off plastic explosives, which require an electric detonator to explode. It is also very unlikely that a semtex explosion could be mistaken for a gas mains explosion (by an expert), as they have entirely different profiles.
At the planetarium the narrator says that the Earth would fit inside Jupiter 11 times. The radius of Jupiter is about 11 times that of Earth, but the volume is 11^3 or ~1300x bigger (the number of Earth-sized planets that would fit inside of it).
When Sherlock talks about Clostridium botulinum, and the botulism toxin, the cutaway shot towards a picture of the organisms is incorrect. Under a normal optical bench-top microscope you would not get false coloring like that shown, and C. botulinum is a Gram positive spore-forming rod bacteria. What is shown is not C. botulinum. Also, the neurotoxin that causes botulism can easily be detected during an autopsy through clinical presentation and serology. Death would not occur as quickly by the method of administration described.
The Museum curator's name is said to be Wenceslas, which Sherlock thinks is Czech. Wenceslas, however, is an English transliteration of the Czech name Václav (and it would have been Václavová for a woman). Additionally, Wenceslas/Václav is a first name, not a surname. The name was possibly chosen for its connection specifically to Bohemia, rather than the Czech Republic as a whole.
Hanging is not a method of execution practiced in Belarus. The only method of execution used is execution by shooting.
When on the case of Connie Prince, the TV shows "Make-over Queen Connie Prince died at 48". DI Lestrade then says she was 54. Apparently she was lying about her age.
There is a view of the London sky filled with stars. Because of light pollution there is no way one can see that many stars in a major city.
At the beginning Sherlock is smugly correcting a prisoner's use of past and present tense. In particular, "I WEREN'T a real man" to "I WASN'T a real man".
However, at the end, Sherlock incorrectly asks the question "What if I WAS to shoot you now? Right now?". Which, of course should be what if I WERE to shoot you right now" [the subjunctive].
In the beginning of the episode, the time that passes between Dr Watson leaving his and Sherlock's apartment and the explosion across 221b Baker Street, is less than a minute. However, Dr Watson only hears about the incident the next morning. He couldn't have gotten so far away from Baker Street in so short a time that he didn't hear the explosion when it happened.
After solving the 3rd case, the bomber shots and blows up the blind woman when she begins to tell Sherlock about his(the bomber's) voice. Sherlock even says in the next scene that he killed her because she started to describe him. However as soon as the woman was found she would have described him to the police.
Shortly after repeatedly correcting a man's grammar, Sherlock says: "It doesn't matter to me who's Prime Minister, or who's sleeping with who", so making two mistakes in one sentence. Someone so fussy about the grammar would have said: "It doesn't matter to me who the Prime Minister is, or who is sleeping with whom".