Continuity mistake: When Buffy and Willow are talking about the sham of Thanksgiving Day at the groundbreaking ceremony, on one angle (the one where Buffy's face is shown) there is a strong wind blowing, moving their hair about, but in the other (when you can see Willow's face) their hair aren't moving.
Continuity mistake: In 'Gone', when invisible Buffy is typing on the social worker's keyboard the letter 'B' is pressed several times. When she prints what she's been typing seconds later it's revealed to be "All work and no play make Doris a dull girl", a phrase which contains no 'B's.
Continuity mistake: In 'Graduation Day, Part 1' when Buffy and Faith are fighting Faith falls into a table, knocking all the comics on it onto the floor. In one or two later shots you can see (on the very edge of the screen) that there's still a comic on it.
Continuity mistake: In 'Becoming, part 1' as Buffy kneels over Kendra's corpse you can see the door over her shoulder, it stays closed all the time. There is a cut to a shot of Buffy from behind showing a cop raising his gun, but he couldn't have come through the door without us seeing him in the last shot.
Other mistake: When Angel, Drusilla and The Judge enter the mall, The Judge kills a pedestrian as he tries to get away. But no other pedestrians that are in the mall notice - A big blue guy just set a man on fire and disintegrated him into tiny pieces in a public mall and NOBODY noticed?
Plot hole: When Dawn and Buffy discover Chloe hanging in the bathroom, Dawn gives a little gasp but nothing more. But, seconds later, some Potentials run from across the hall saying "What is it? We heard -" before they see Chloe. While it suits the next scene to have them there, there wouldn't have been any sounds loud enough for them to have heard from across the hall through a closed door.