Revealing mistake: When Anya sings her song about how she will become Mrs. Harris, she spins Xander's chair around after putting a blanket on him. When she is also sitting on the chair, and you see an overhead shot of them both, you can see a section of the stage floor rotating beneath them keeping the chair spinning.
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.
Chosen answer: From what we see in the series, certain girls are identifiable as potential Slayers - Kendra clearly fell into this category. The identification method is presumably mystical in nature, but the Watchers' Council are pretty effective at that sort of thing, so they're quite good at tracking down the potentials ahead of time. Not perfect, though - it does appear that Buffy herself may have slipped through the net - certainly she had no inkling of what she was until she'd already taken on the role of Slayer. It is possible, however, that this was actually cultural - an American family would hardly be likely to turn over their daughter to some strange man for 'training', so the Watchers might have chosen to keep an eye on her covertly, whereas some other cultures (like Kendra's Jamaican parents) might be more willing to believe.
Tailkinker ★