Continuity mistake: Right after Cordelia has her vision, Wesley goes to the whiteboard. In the "Case" column he writes "N.D.U.O." underneath what looks like "CARYNSS" crossed out. But once they find out that it is a Prio Motu demon and Angel writes it on the whiteboard, "N.D.U.O." disappears even though Angel did not rub it off.
Continuity mistake: After Wesley, Cordelia and Gunn sing "We Are The Champions" and Cordelia mentions tequila being evil, we see Wesley put aside some shot glasses. These glasses change position three times throughout the rest of the scene without anybody touching them.
Continuity mistake: When Angel writes "Prio Motu" on the whiteboard and says that they need to find it and narrow it down to somewhere, Wesley says "I may be able to help us with that." In the immediate closeup of Angel, his stance has suddenly changed. In the previous shot he was standing with his arms straight down and his hands close together, but in the closeup they appear to be as though he has his hands on his hips.
Continuity mistake: In season three episode "Anne" of BTVS, The character "Anne" who later comes to season two of Angel running a youth shelter had a rather large red tattoo of half a heart on her arm. However, no trace of it is to be found when she is on Angel. Tattoo removal is a rather pricey procedure, and she seems to be tight on cash until she gets the two million. Even if W&H would have paid for the procedure, there would be some scarring or residual pigments for such a large tattoo. (It's definitely the same character - Anne says that she used to think vampires were cool, until she met one. That's a reference to her first appearance (as "Chanterelle"), in which she belonged to a vampire-loving cult.)
Revealing mistake: In the scene where Angel fights the demon for the first of his trials, upon cutting the demon in half for the second time he chains the two halves to the walls. When he drags the upper half of the demon to the wall, the green legs of the actor are momentarily visible, when they should have been removed by CGI.