Visible crew/equipment: When Casey tells the killer "Dial somebody else, okay?", you can see the shadow of some equipment on the decorative pillar at the top right of the screen as the camera moves backward. (00:03:18)
Revealing mistake: When Sidney in Ghostface costume stabs Billy with the umbrella, you can see the squared shape of a protection pad under Skeet Ulrich's shirt (which she misses the second time around, famously prompting a genuine scream of pain from the actor). (01:39:55)
Visible crew/equipment: When Casey agrees to play the killer's game she crouches down next to the television and as the killer says "Here's how we play..." you can see a clear shadow of the head of the boom microphone cast on the white wooden railing at the top right of the screen. (00:06:25)
Revealing mistake: After Billy's stabbing, Stu charges Sidney projecting her through a doorway. In that shot you can easily see that Matthew Lillard is replaced by a stuntman. (01:40:15)
Other mistake: In the first scene with Randy telling Jamie Lee Curtis to turn around, the reflection in the TV never mirrors the actual position of Jamie Kennedy. (01:22:30)
Audio problem: Gale exits the TV truck telling her cameraman "I'll be right back" to go with the theme of the movie. However, the line is dubbed over and Courteney Cox says something shorter and different. (01:14:25)
Continuity mistake: When Dewey and Gale enter the house party Dewey jokingly takes a bottle of beer off a party goer then gives him it back. The bottle is back in Dewey's hand again when he speaks to Tatum. (01:03:40)
Revealing mistake: When Sidney is attacked by the killer for the first time (in her house), she is knocked to the floor and you can see she is played by a stunt double. (00:28:39)
Continuity mistake: After Casey has been stabbed in the chest she falls down and the killer gets on top of her, before he can stab her again she knocks the knife out of his hand and the killer instead starts to strangle her. She then kicks him off her but you can see he still has the knife in one of his hands as he falls backwards. (00:11:00)
Continuity mistake: In the final scene where Sidney gives the "coup de grace" to Billy shooting him in the head after he suddenly woke up, he holds the left arm above the mask on his top left, but when he dies, his arm is lowered and the knife appears from nowhere. (01:42:30)
Continuity mistake: After her first time with Billy, Sid brushes her hair. She puts the brush back on the bedstand, but the phone on it changes angle between shots. Later in the scene, you can also see the brush's handle sticking out from the side of the bedstand by the bed, or the one facing front. (01:18:55)
Revealing mistake: When Sidney discovers Tatum's body, we see her head intact, in particular the eye level. In her death scene, her cranium was being crushed right at the eyesocket. (01:07:25 - 01:22:15)
Continuity mistake: Mr. Himbry calls the two expelled students "Desensitized little sh*ts", pointing the open scissors at him. In the next shot he's looking at him but he changed position, since the scissors are gone from frame. (00:45:20)
Continuity mistake: Gale falls back on the unconscious Dewey after Billy kicks her. Her jacket is draped across the chest touching the white button of her vest. When Billy gets on his knees by her though, the jacket is fully open. (01:37:30)
Continuity mistake: Once Billy is done slashing his accomplice, he points the knife at him and tells him what to do; the knife hand keeps changing position between shots. (01:36:00)
Continuity mistake: Plummeting from the roof onto a boat, Sid causes a wetsuit (or a similar kind of fabric) to fall off. In the closer shot, the cloth is much more entangled than in the other view - look at the pink band. (01:21:55)
Continuity mistake: Off-balance on the roof as she shouts for help, Sid is hanging by the window with one hand or both depending on the shot. (01:21:45)
Character mistake: Sid deduces that Billy could be the killer because he could have used his phone call from jail to make the threatening call. However, the deduction has to be incorrect, since if it was really Billy, he could have never used the voice-changing device.
Suggested correction: This will sound pedantic (hey, it's MovieMistakes.com, we're entitled), but I don't think this is what is meant by a character mistake. Sidney isn't presenting an erroneous fact that goes uncorrected, she's explaining an idea that had crossed her mind. Billy immediately gives a plausible response and by the end of the film it's evident who'd called her. The takeaway isn't that Billy could still be the killer but that Sidney can't help but suspect him despite their reconciliation.
Other mistake: Tumbling down the ditch with Dewey, you can see that Gale loses her red pumps. In the rest of the scene though she goes to Mr. Prescott's car first and then back towards the street without fetching her shoes back. (01:17:55)
Continuity mistake: At the party, Randy asks (with a weird "yeeeeh!" that appears to be dubbed over) how many "hellraisers" are there. Matthew Lillard points at Rose McGowan two times in two different shots and not with the same hand either. (01:02:55)
