Welcome To The Hellmouth (1) - S1-E1
Visible crew/equipment: When Buffy's mom drops her off at school at the beginning, watch closely when Buffy gets out of the car. In the passenger door's quarter glass window, you can see a reflection of a set-light (probably a fill light to eliminate shadows) for a few seconds when she opens the door.
Welcome To The Hellmouth (1) - S1-E1
Visible crew/equipment: In the beginning, when Buffy is having the prophetic dream, there's a shot right as it's about to end where the camera is above her and moving down closer to her. If you watch, you can blatantly see a shadow (more than likely the camera rig) grow over her as it gets closer and closer to her.
Visible crew/equipment: When Catherine and Amy get switched back to their original selves, Catherine takes the fire axe that Amy is holding by using her powers. The wire used to pull the fire axe towards her as Amy holds it up and says "Mom! Please!" is visible.
Never Kill A Boy On The First Date - S1-E5
Visible crew/equipment: When Buffy and Giles are in the cemetery, right before Buffy bails to go meet Owen, you can blatantly see a number rectangular studio/set lights reflected in Giles' glasses coming from various angles. (There wouldn't be dramatically-angled lights like that in a cemetery, let alone after dark).
Visible crew/equipment: When Buffy, Willow, Xander and Giles all agree that Morgan is their one suspect, a camera and a cameraman can be seen in the reflection of the glass in the door labeled "Librarian", right after Buffy says, "Uh, priority check, Giles." A boom mic is also reflected in the top right of the glass when Buffy finishes her line with "talent show murder!" The mic can be seen throughout the rest of the scene from two different camera angles.
Visible crew/equipment: When Buffy is searching through the backstage area (before the chandelier falls on her), there is a mirror in the background. At one point in the mirror, you can see the reflection of a crew member. Pause for a better look but it can be seen at normal speed.
Out Of Mind, Out Of Sight - S1-E11
Visible crew/equipment: When Buffy and Cordelia enter the mop closet you can see the shadow of the boom microphone on the wall near Buffy's head (left side of screen). (00:31:44)
Out Of Mind, Out Of Sight - S1-E11
Visible crew/equipment: At the very end, as the invisible girl gets taken along a corridor by the FBI agents, check the top of the screen - a boom mike creeps in as the agent says his line, then is withdrawn.
Out Of Mind, Out Of Sight - S1-E11
Visible crew/equipment: After Cordelia wins May Queen, Buffy, Xander and Willow are talking, and right before she walks away, you can blatantly see an on-set light reflector being reflected in Buffy's sunglasses.
Out Of Mind, Out Of Sight - S1-E11
Visible crew/equipment: When Giles, Willow and Xander are trapped in the storage room with the leaking gas, at one point you can blatantly see a massive set-light reflected in Giles glasses. (This is fairly common, as often lights are reflected in his glasses, but it's particularly noticeable here).
Out Of Mind, Out Of Sight - S1-E11
Visible crew/equipment: When Angel and Giles talk in the library, in multiple shots you can see stage lights and light reflectors being reflected in Giles' glasses. (This is actually fairly common in many of his scenes, but it's particularly noticeable here).
Visible crew/equipment: When Buffy meets Giles in the library after the opening credits, and they start talking, throughout the scene, you can see set lights and reflectors reflected in Giles' glasses multiple times. This happens often throughout the series, but is particularly noticeable here.
Some Assembly Required - S2-E2
Visible crew/equipment: As Buffy goes through the door with the "Keep Out" sign on it, the shadow of a camera can be seen on the left side of the door in the last few frames of the shot.
Visible crew/equipment: When Ampata agrees to go to the dance with Xander and they walk off down the hall, a shoe can be seen in the bottom right of the screen within the last few frames stepping into shot.
Visible crew/equipment: Right after Kane and Buffy's fight in the woods, in the shot right after Kane leaves, you can see set-lights above the clearing where they fought. (00:39:10)
I Only Have Eyes For You - S2-E19
Visible crew/equipment: When Xander opens his locker and the gross hand grabs him, if you use slow-mo when the shot is inside the locker looking out, you can see the white t-shirt of the man whose arm it is. (00:10:20)
I Only Have Eyes For You - S2-E19
Visible crew/equipment: After the history teacher writes, "Don't walk away from me, bitch!" on the chalk board, Buffy says to Xander, "I'm telling you, something weird is going on." As she says "something", a boom mic is reflected in the window behind her on the right.
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 ★