Revealing mistake: In the shot right before Coraline finds the hidden door for the first time, she says "Alright Little Me, where are you hiding?" and drops her notebook and pen on the floor. When the notebook and pen are falling, you can see the wires that were used to suspend them while they were being animated. It's a common stop motion technique, but these days they usually try to paint the wires out afterward. It's a pretty severe dutch angle though, and they're only visible for a few frames, so it's easy to miss.
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.