Revealing mistake: In "To Shanshu in LA" in season one, as Angel is pulling out of parking lot to follow the W&H Trucks, you can see his reflection in his rear view mirror (vampires have no reflections). Also odd that throughout the first season the rear view mirror is removed in some episodes. (00:31:19)
Factual error: According to the show, Angel learned quite a few languages when he was evil. However, when he tries to start a conversation with a red-faced demon, he makes terrible mistakes. He said "Hablo Espanol?" which literally means "Do I speak Spanish?" and he said something like "Are you from Byelorussia?" in a tongue that hardly resembled Russian, although according to the captions, it was said in Russian.
Continuity mistake: When Gunn is talking to Wesley near the start of the episode, he mentions that he's tracked Lindsay down, and that he's been living under the name "Doyle", and holds out the lease of his apartment. There's a close-up of the lease a few seconds later, in which the name "Lindsay McDonald" can be seen.
Continuity mistake: In Season One Finale, "To Shanshu in LA" when Wes is lecturing Cordy on wants and desires, the number of bites taken from the doughnut changes, in one instance it breaks in half, most visibly when Wes takes a bite from Cordy's doughnut on the angle change. (00:13:31)
Continuity mistake: In the scene where Angel has just returned and given Cordelia her necklace, watch Cordelia's hands; they keep changing position in each shot. In one shot she is trying to put the necklace on, in the next shot she appears to have given up and brings her hands down and bundles the necklace in her hand. Then when the shot changes to behind her, she is trying to fasten the necklace behind her neck again. This happens a few times during this scene.
Factual error: Wesley has his throat cut, deeply enough to at least nick the artery (based on the amount of blood) and the windpipe (based on his difficulties breathing and talking later on), yet he somehow survives being left alone in the park for over 24 hours. In reality, he would have been dead long before then.
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)