Continuity mistake: At the beginning, the way Holmes holds the photo is different in consecutive shots. (00:01:00)
Character mistake: The American Flag is hung improperly on the wall. The Flag's union (stars) should be at the top, to the observer's left, but it's to the observer's right. (00:06:38)
An Unnatural Arrangement - S2-E6
Continuity mistake: At the start, when The Woman first fires at the door the holes are quite dark and splintered. Then when she moves to pick up the phone the holes look far neater, like they've been drilled.
Chosen answer: M broke into the brownstone and left a note. Holmes deduced he was staying at a high-class hotel from the soap scent on the note, and gave a picture from a hidden camera to his "irregulars" on the street to stake out hotels. One of them saw him, so Holmes followed him to his next intended victim and knocked him unconscious with a baton.