Visible crew/equipment: At the end of the episode where Wesley is studying the Shanshu scroll in Cordillia's apartment, the boom mike appears at the top of the frame TWICE.
Revealing mistake: When the insect creature is crawling on the ceiling its hair doesn't falls down towards the floor, it "falls" towards the ceiling. (00:22:05)
Continuity mistake: When Cordelia is putting on lipstick she uses the windows as mirror. In the reflection we see the top drawer of the cabinet is open the whole time, but it is later opened by Angel. (00:00:10)
Continuity mistake: In the season one finale "To Shanshu in LA," near the end Angel is fighting Vocah in the mausoleum and Angel rips off Vocah's mask, revealing the maggoty face Underneath. In subsequent shots of Angel from behind Vocah, you can still see the edges of the mask on his face. (00:34:30)
Continuity mistake: In the final fight, Angel gets stabbed on his right side. In the slow-motion bit before his opponent chooses not to fight any more, the wound switches to his left for one shot. (00:36:40)
Character mistake: When Faith arrives in LA she is carrying a bag. She puts it on the ground and then steals a man's jacket and wallet. Afterwards however, she leaves without picking up her bag.
Continuity mistake: In the scene where Detective Lockely arrests Angel in the apartment, the window behind them shows that it's daytime. But when Angel jumps out the window, it's night time.
Continuity mistake: When Faith is sitting at the window and looks at the piece of broken glass she used to torture Wesley, it has a different shape than the one she first takes to torture Wesley. When she drops it out the window it has its original shape again. (00:33:10)
Deliberate mistake: In the flashback to Angelus rising from his grave, his and Darla's breath is visible as they exhale. This was going to be removed in post-production, but left in due to budget restrictions.
Character mistake: When the bidding for Cordelia in the auction reaches $20,000, somebody bids for $30,000 after the auctioneer has said "going once, going twice" and they end up winning. The auctioneer should have in fact restarted saying "going once, going twice" after the $30,000 bid was made.
Blind Date - S1-E21
Character mistake: Angel comments that Vanessa was acquitted because of a hung jury. A hung jury means the jury couldn't come to a unanimous decision; the case would require a new trial, while an acquittal means the jury found her not guilty, and she couldn't be tried again for the crime.
Plot hole: During the scene where Angel tries to work out where Wesley is being held hostage we learn that the man Faith beat up at the start is still alive in the hospital and that it's his apartment she's staying in. Angel bursts into the apartment to save the day, but that's not possible without an invitation, as the owner is alive, whether or not he's actually there. (00:34:00 - 00:36:40)
Continuity mistake: When Wesley is explaining that Angel feels like he doesn't belong in the world, the amount of donut he's holding changes. (00:13:30)
Visible crew/equipment: When Angel is fighting the demon a cameraman is visible rising on the lower right. (00:27:13)
Visible crew/equipment: When Angel kneels down to the fridge a camera lens is visible in the top left. (00:15:21)
Visible crew/equipment: When Angel is drop kicked to the floor a camera LCD in the lower right is visible. (00:24:35)
Revealing mistake: At the vampires' nest, when Knox is talking about Gunn's team, the vampire wearing the green sweatshirt and leather vest doesn't have yellow eyes like the other vampires do. And the female vampire doesn't either later, when the two groups face off after Gunn's sister is sired. (00:22:30 - 00:35:35)
Answer: Everybody including Angel saw her. After helping them and then being left alone to talk to Angel, Cordelia says that the Powers That Be let her come back one more time to help them. After Angel gets a call saying that she died, he realises that it was her spirit that was helping them the entire time.