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.
Character mistake: While the original Sherlock Holmes is famous for making water-tight deductions, the one in this episode does not quite live up to that standard. He declares someone a "compulsive cheat," because: "Waterproof cover on his smartphone. Yet his complexion doesn't indicate outdoor work. It suggests he's in the habit of taking his phone into the shower with him, which means he often receives texts and emails he'd rather went unseen." There are far too many problems here. (1) The only type of complexion that indicates outdoor work is sunburned. We don't have water-soaked or snow-frozen complexions. So, yes, he might work outdoors. (2) Indoor work may also deal with water, e.g. plumbing, dish washing (note what the accused man is doing!), water purification, lab work, etc. (3) Some people might just purchase a waterproof cover as a precaution. Some might receive it as a gift. Some may receive it free during a bulk purchase. (4) Not everyone who takes his or her phone into the shower has fidelity problems. They might want to listen to music, be up-to-date with the stock market, or, if I may use a euphemism, might want to have some quality alone time. (00:12:35)
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.
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)