Plot hole: Spike must have had his chip implanted while he was unconcious. If so, how did he attack The Initiative scientists without any pain when he tried to escape? You could argue that they programmed him to be capable of violence within The Initiative only, but why would super smart scientists program an evil being to be able to escape at the first opportunity? Joss Whedon has acknowledged this as an oversight.
Plot hole: Willow hacks into The Initiative's computer system and discovers air ducts and electrical conduits leading to Adam behind 314, but the room doesn't show up on the schematic. It seems unlikely that nobody who didn't know about Adam would have noticed this and investigated.
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 ★