Continuity mistake: The underground car seen in the CCTV video is from the Jubilee line, but is meant to be the District Line. It's also different from the one where the heroes defuse the bomb, which is an actual District Line train (D stock). The rounded doors is the most visible proof.
Factual error: Sherlock takes pains to explain in detail that he will deliberately corrupt the magnetic code stripe on an access control card he intends to use to enter the elevator to the villain's penthouse, by carrying it next to an operational cell phone. (This is possible due to the low frequency magnetic field from a phone's vibration motor.) But when he actually goes to access the elevator, he simply touches his access card to the reader instead of swiping or inserting the card through a slot, which is how a magnetic stripe reader would operate. The elevator uses an RFID proximity reader, not a magnetic stripe reader - a phone wouldn't corrupt an embedded RFID tag.
Revealing mistake: Before John Watson closes the laptop, the screen displays a phone tracking website in the Chrome web browser, supposedly at Mephone.org.uk. In the URL address bar, however, "file:///C/http/mephone.org/mephone.html" is visible which is a standalone HTML file on the computer's hard drive. (01:08:35)
Revealing mistake: The banker shoots himself with a Sig 226R. When you see it lying on the bed it has a screw missing from the grip. In the first episode, Watson has a gun in his desk drawer. It is a Sig 226R also with a screw missing from the grip. Obviously they only have one prop gun.
Plot hole: Sherlock manages to convince Janine to let him into Magnussen's quarters after deceiving her into thinking he is going to propose. Several days later, she is enjoying her "revenge with profits" on Sherlock for his duplicity. Problem: There is no way she could have found out. She was out cold moments after letting Sherlock in, and Sherlock himself was shot by an assailant, which he had misidentified. Only Watson knew Sherlock's true intention, but we know he did not reveal it. The shooter and Magnussen did not know and had no opportunity to tell Janine anyway. (Plus, Magnussen's credibility is too low.) In fact, if I were the director, I wouldn't bother making up a motive for being a "grasping, opportunistic, publicity-hungry, tabloid whore" (film's own quote). Such a person loves "profits", not just "revenge with profits." (00:31:00 - 00:43:00)
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)
Continuity mistake: When Sherlock steps out of the taxi on Baker Street three extras walk by. First, it's a lady with her back to the camera, wearing a dark jacket, purse, boots and a dark curly hair. Next, the two women walking towards the camera. Notice the blonde one on the left with the pony tails, dark jacket with the multi colored cloth at the bottom, and grey pants. Fast forward to when the taxi pulls up to take Sherlock and Watson to the the crime scene. As Sherlock opens the taxi door, the same first lady with the curly hair and the blonde with the ponytail are now walking together in the same direction the blonde was walking when Sherlock arrived. (00:12:58 - 00:17:51)
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.