Audio problem: When Willow and Jenny jump in Cordy's car, Cordy mentions that she saw "these things coming at me!" However, her mouth isn't moving for the last few words.
Deliberate mistake: When Willow and Jenny go out to the parking lot only to discover vampires converging on the school, pay attention to the extras. You can make out some of the same extras in different shots coming from different directions. (Most notably, you can see a woman in a long black-and-white skirt coming from the direction of the parking lot, then a few shots later, she's suddenly over in the football field coming from there.) They obviously reused extras in different shots to "pad out" the crowds.
Continuity mistake: When Buffy descends into the Master's lair, the way she is holding her crossbow changes depending on the camera angle. Sometimes it's pointed straight forward, sometimes it's pointed upwards.
Continuity mistake: When Giles and Jenny are talking about Buffy, Giles says "We must get in touch with this Brother Luca chap!" watch Giles. In the first shot he's still reaching towards the top of the cabinet. Instantly in the next shot, he's in the middle pulling something down from the top of the cabinet.
Continuity mistake: When Giles and Jenny are talking in the library, when Jenny closes the bible, the way she's holding it changes instantly between shots.
Continuity mistake: When Buffy is consoling Willow after she discovered the dead students, as Buffy leaves the room, Willow's mouth is open and she looks down. It then cuts to a wideshot. Instantly, Willow's mouth is closed and she's staring forward, and she looks down a second time.
Continuity mistake: When Jenny meets Giles after Xander's botched attempt at asking out Buffy, the way she's holding the folders to her chest subtly changes repeatedly between the closeup and wide shots.
Continuity mistake: When Buffy turns down Xander's date proposal at the start of the episode, throughout the scene, Xander has his backpack up around his right shoulder. However, when he walks away, it's instantly down by his side.
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.
Plot hole: Angel says he cannot give Buffy mouth-to-mouth because vampires have "no breath." However, throughout the entire series, vampires are shown to breathe, and are able to do things like smoke, gasp, get choked out, etc. (Granted, the franchise also confirms a vampire cannot die from asphyxiation, but still they're shown to breathe regularly).
Continuity mistake: After Buffy finds Morgan's brain, Willow pulls up Morgan's school file on one of the library computers without even touching the keyboard. One moment the monitor is on the desktop, the next it's showing Morgan's file.
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.
Continuity mistake: In the episode 'Teacher's Pet', when Ms. French is eating her bug sandwich, her sleeves go from being up in the long shots to being pulled down in the close-up of the sandwich. (00:24:05)
Welcome To The Hellmouth (1) - S1-E1
Continuity mistake: When Giles is giving his speech about the Hellmouth the books Buffy's holding keep changing from facing him to facing Buffy and back again. (00:18:20)
Continuity mistake: Buffy's birthdate starts at 1979. In "I Robot, You Jane", when the demon pulls up Buffy's file in the school system, it shows her year of birth as 1980 and her year as sophomore. But when her file comes up on the student's computer screen, it's changed to 1979 and that she is a senior. It wasn't until towards the end of season 1 in "Nightmares" that her birth year is 1981 (on her tombstone before she comes out as a vampire). (00:08:30)
Never Kill A Boy On The First Date - S1-E5
Other mistake: In the famous scene where Buffy tells Giles, "If the Apocalypse comes, beep me," she grabs her pager from out of shot. Trouble is, she's standing in the middle of her hall and there's nowhere she could have picked the gadget up from. (00:21:00)
Revealing mistake: When Sid stabs Marc after he has transformed, there is no blood on the knife that he uses afterwards.
Chosen answer: "So goes the nation" seems to have been used on many occasions, with various different US states in the "As .... goes" section. Most commonly it seems to be California that's considered to lead the way, but probably most other states have appeared in the lead role at some point or another. Other things have also been used - no less a person that Pope John Paul II said "As the family goes, so goes the nation...". The origin of the quote format is unclear - in US politics it goes back into the 19th century, when it was Maine that held the title spot, but, while no definitive origin is known, it seems highly likely that it goes back considerably further than that.