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)

Angel (1999)
2 plot holes in season 1 - chronological order
Visible crew/equipment: When Angel is fighting the demon, you can see an obvious camera and two hands in the bottom right corner. (00:27:10)
Cordelia: You want I should distract him? Make with the nice-nice while you slip by?
Angel: Don't be stupid. I'm that guy and the most beautiful woman I've ever seen is making eyes at me? It's either a bachelor party or a scam.
Cordelia: What did you just call me?
Angel: I'm sorry. You're not stupid.
Cordelia: No, after that.
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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.