Continuity mistake: During the other father's song, notice the red record player by the door and the position it's in, with the lid facing Coraline when he says, "making up a song". When the camera angle changes and he says, "she's a doll", the record player's position changes completely, with the lid now facing the wall.
Continuity mistake: During Coraline's first meal with her Other Mother and Father, just before Coraline says "I'm real thirsty." there are two drinking glasses to her right, the one she uses a couple seconds later and a smaller one. After the cut to the chandelier lowering, it goes back to Coraline and the smaller glass has completely vanished.
Continuity mistake: When Coraline goes to see her father for the first time in his study, there are three boxes at the entrance. She enters, puts the Coraline doll down on the smallest box with the doll's legs and right arm hanging down, then she starts playing with the creaky door. When her father tells her to stop, the boxes have changed position and the doll's legs and arm are straight.