Continuity mistake: When the detective is interviewing the woman in the alley, he mentions that the murder took place "last night". In "Bad Girls", we see Buffy visiting Faith the day after the murder, and that night is presumably when she has the dream about drowning, so by the time the detective is asking this question it is actually two nights after the murder, not one.
Revealing mistake: When Spike tells his mother that he's become a vampire, she asks him if he's drunk. He responds with "Little bit." Just after he says this, someone can be heard coughing off-camera. It could be his mother because she isn't supposed to be well, but it seems odd that the camera wouldn't be on her during this. And a cough wouldn't really be necessary for the scene.
Revealing mistake: When Spike is talking to the Anointed One for the first time, one of the henchmen tries to rush him from behind and Spike backhands him without even looking, but Spike's hand movements, the reaction of the henchman and the sound of the punch connecting on the audio track are all out of sync.
Continuity mistake: When Wesley and Cordelia kiss, after a few seconds they freeze in place. Cordelia has her left arm around his waist, we see a shot of her face for about a second with her arm definitely angled downwards, then we cut to Wes' face and her hand's now on his neck.
Visible crew/equipment: During the scene of the siege at Giles', there is a shot where an Indian guy jumps through the window and helps himself by holding on to the sill. If you look to the left side of the screen, you can see it's a moving wall and that several hands of crew members are holding it so it doesn't shake.