Although "A Scandal in Belgravia" is the fourth Sherlock episode, I begin with my review of it because (a) it is so very popular and (b) it is so appallingly bad.
Yes, I know it won the Edgar (proving again, as if any more proof were needed, that awards are for popularity and not for merit). That doesn't change the reality that it is not only the worst Sherlock episode (at least, by the end of series 3), but a bad episode in general. Steven Moffat is a professional writer; he has no excuse for pumping out this disaster. That he is actually getting patted on the back for it is intolerable.
Granted, the incidentals are probably the best in all of Sherlock. A pantless visit to Buckingham Palace; Sherlock's expecting John to hear him when he isn't even in the country, let alone the room; Sherlock's "not remembering" the number of times he threw the CIA man out the window; and (my sister's favorite) "Vatican cameos!" shine in an already luminescent series. "Belgravia" can't be faulted for its entertaining trappings, but like misdirection in a conjurer's illusion, they hide the very poor essentials of the piece.
These problems are so very numerous that I do not believe that the average Internet reader will plow through a single article on them all. Therefore, I have split my article into bite-size chunks, each covering a different flaw in the writing of the episode.