Continuity mistake: We see John come into the room and stand next to Mrs. Hudson and put on his coat. After a few shots Mrs. Hudson walks away with the plates and says, "Lovely tune, Sherlock..." As she's walking away we see John isn't wearing his coat, and he puts it on again. (00:50:00)
Other mistake: While Sherlock sits reading the paper he gets a text alert from (we assume) Irene Adler, and John notes that he had heard this alert fifty seven times. At the end of the episode, when Sherlock is looking through the texts that she sent to him, there are not that many texts from Miss Adler.