Visible crew/equipment: When Xander is in the bathroom after Buffy called him to look for the talisman, he bends down and grabs the talisman that lays under the sinks. As he stands the camera pans up to the mirrors and on the left side you can see the reflection of a crew member crouching over the hole to the old school. It's not the female manifest spirit as the clothes are wrong and she then appears from a different part of the room.
Storyteller - S7-E16
Visible crew/equipment: During Andrew's fantasy of him, Warren and Jonathan being Gods, something dark is reflected in of the piles of gold. Possibly a camera or camera crew.
Continuity mistake: When Willow is out with Kennedy, Willow tells her that she doesn't get why she likes her. As she says "I still don't get it", she pulls her hair back over her left ear. Then when she says "Why you like me" in the following shot, her hair is hanging back down over her left ear again.
Lies My Parents Told Me - S7-E17
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.
Dirty Girls - S7-E18
Revealing mistake: In the opening scene, where Caleb picks up the potential slayer in his truck, there is a brief shot of the two from the hood that pans slightly to the right. In doing so it reveals the edge of a screen showing a fence and trees passing by, revealing that the car is in fact stationary and in a studio. (00:02:40)
Visible crew/equipment: When a guy asks Dawn out to Winter Formal and she says "What?" - you can see a microphone at the top of the screen. (00:33:35)
Continuity mistake: When Buffy is giving her speech about being chosen, there is a shot of Andrew sitting on the couch next to Xander and Anya, but in several of the shots of the Potentials he can be seen standing on the other side of the room in front of the curtains. It's definitely him - his face isn't visible, but the clothes are the exact same.
Get It Done - S7-E15
Continuity mistake: In the episode 'Get It Done' when Dawn and Buffy find Chloe's corpse the door across the hall from them is open. When it cuts back to them after the ad break it is shut, but there has been no time lapse.
Visible crew/equipment: After Buffy has spoken with Clem and is walking down the road, you can see the reflection of a camera in the back of a green SUV.
Other mistake: We're shown a variety of photographs and pictures of different members of the gang at one point. One of those, the one of Xander and Anya, is actually a shot from the episode "Something Blue". I doubt very much that during the events in that episode, someone conveniently decided to take a photograph.
Lies My Parents Told Me - S7-E17
Other mistake: Robin uses the song that Spike's mother sang to him to provoke him into attacking him. The song is played on his computer through Apple's media program "iTunes". He clicks his computer's mouse to start the song, but when we see the monitor the cursor isn't placed on the Play button. Robin doesn't jostle the mouse when he presses it, it remains completely still on the desk. And the only way to play a song through iTunes using a single mouse-click is by clicking on the Play button.
Continuity mistake: In the very last episode, Chosen, in the shot right before Spike's amulet sends out the blast through the roof, you can see an ubervamp just about to pin him against the wall and in the next shot he is standing in the middle of the area backing up toward the stairs and then he sends out the blast. There would never be enough time between the shots for Spike to break free and then go and stand in the middle of the floor.
Storyteller - S7-E16
Audio problem: After Anya and Xander sleep together again, Anya says "I think, maybe we're really over. Which - it's good, right? I mean, now we can move on." When she says "Which - it's good", her lips don't match what she's saying.
Continuity mistake: When Buffy ties Spike to a chair before the opening credits, she ties a rope around his left arm three times. But two extremely fast shots later, the rope is tied around his left arm four times.
Continuity mistake: In the flashback to St. Petersburg, 1905, just after the burning man runs out of the room, Halfrek mentions the Revolution. Between shots, Anya's champagne glass switches from her left hand to on the table and back.
Answer: She has superpowers, ergo fast movement. The reason she outruns him so easily in "Touched" is that she stops trying to actually fight him and just dodges instead.
Shay