In the classroom, Casper ties Amber's shoelaces together before anyone else's. When Amber stands up to protest the party being at Kat's house instead of hers, she doesn't trip over them. She sits back down, but when the bell rings and she gets up, she trips.
When Carrigan is walking down the hall with the axe in her hands looking for Dibbs, the bottle of potion is nowhere to be seen. When she goes through the window, the bottle is lying next to her on the ground.
When Casper is pouring the fresh orange juice for Kat, he collects the pulp in his hand. When he tips the pitcher up, the orange juice pouring through his hand stops just before he tips the pitcher back.
When Dr. Harvey rolls down the stairs, he gets wrapped in the carpet on the stairs. When he gets to the bottom of the stairs he keeps rolling, and the carpet unrolls itself onto the floor. He going in the wrong direction for that to happen.
When Casper grabs Kat's arm and drags her toward the window out of the room, she leans back and gets dragged on her heels, thus pulling her arms stiff straight out in front of her. Before they're out of the room, he starts to lift her in the air but Casper is still in front of her and holding her arms stretched out in front of her. He should have been above her head and her arms stretched out above her.
Ghosts don't have a reflection. When Casper looks into a mirror, he can't see himself. But when Amber and Vic are dressed in costume for the party, Stretch, Fatso, and Stinkie spook them, and appear in the mirror. However, these are not their reflections. When the two kids fall, the ghosts are nowhere around, suggesting that the Ghostly Trio is actually inside the mirror.
Casper tells Kat that after death, life doesn't matter so much anymore, so things tend to be forgotten. Every time a person becomes a ghost, they forget who they used to be and what their purpose was. When Dr. Harvey dies, he can't remember his own daughter. Yet, after Carrigan dies, she still remembers her partner and her duty (to get inside the vault). The ghosts in the movie regain their memory quickly. Carrigan may have remembered her role as soon as she saw Dibs.
When Kat is dancing with Casper at the end of the film, the duo is shown floating above the floor. Yet, in the next shot, a view from the balcony, they are back on the floor.
However, during a shot in between while he's identifying himself to her a light in the background can be seen getting bigger indicating that he's lowering them to the ground.
Casper says that his favorite baseball player is Duke Snider. Casper lived in the late 19th century, and it is assumed that his father died in the 1910s. Duke was born in 1926 and debuted in 1947. There's no way Casper buried the glove and baseball for his pirate games with his dad.
It is clear that the ignition must be turned upwards to remove the resurrection liquid, as shown by Dibs when he stole it. But when Carrigan is turned into a ghost and he refuses to make her live again, he pushes the liquid out of the ignition with it turned downwards. That gesture should have caused the opening of the bottle or the destruction of the ignition. The bottle does not open, but the ignition sparks. That shows it is broken. How they brought Dr. Harvey back to life if the ignition is broken?