When Coraline first looks at the picture frame next to her bed, the "3" from "at 7:30" is missing and "Coraline" is spelled "Coral1n3" on the depicted sign. In the next shot, when she holds the frame in her hands, it says "7:30" and "Coral1ne".
When Coraline first meets the Other Father in the study and is playing his piano, he sings Coraline a song. Notice the red record player in the background is facing Coraline. But when he sings "She's a doll", it is now facing the wall.
While Coraline is eating dinner with her "other parents" she has an empty glass and a full water glass. When Coraline is shown again the water glass is half empty and again when Coraline is shown there is no water glass present at all.
Watch very closely during the opening as the hands unstuff the doll, and restuff it. When you see the doll inside out in full, look down at her feet, with the heel stitches on the same side of the body as the face.
The goof item below may give away important plot points.
After Coraline acquires the last of the eyes of the ghost children and the Other World begins to disintegrate around her, the camera quickly moves up to reveal everything around the house is vanishing. In the beginning of that shot, we can plainly see that the metal stairs leading up to the attic apartment of the Other Bobinsky are missing and rightfully back in the very next shot.