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Art Evolution: Casper's original design was a much chubbier duckpin like shape and had more resemblance to a bedsheet ghost, whereas his later Harveytoons redesign streamlined him into the simpler, big-headed ghost design that we recognize today.
While not exactly bedsheeted, the ghosts are invariably transparent, pale and not at all detailed.
In at least one of the cartoons, many of the ghosts masquerade as sheets in a haunted house, going so far as to cover furniture and serve as pillowcases.
Characterization Marches On: In early Casper comics, he actually scared people. But he eventually got bored and wants to make friends. Though in every other incarnations, Casper is apparently friendly from the start.
Crossover: Spunky the Donkey, who originally appeared in several of the Fleischer StudiosColor Classics and a Famous Noveltoon, appears (curiously, without his mother, Hunky) in at least two of Casper's shorts, over a decade after his previous roles.
Embarrassing Last Name: Casper's last name (yes, he has a last name) is McFadden. Try saying that without laughing.
Felony Misdemeanor: From the way Casper treats it, you'd think scaring people was ax murder or something. Wendy's the same with her aunts, though sometimes they did go farther with their spells than harmless pranks and scares (How does turning a butterfly - remember, all animals are sentient in this series - into a hole in a plank of wood even work?)
Jerkass: The Ghostly Trio. There have even been instances where even Spooky was a little appalled at their schemes, and he idolizes them.
Just Eat Gilligan: If Spooky would just lose the hat that never turns invisible with him, he'd have a much easier time. Of course, the one time he did try to ditch it, nobody recognized him without it, and neither human haunting victims nor other ghosts who wanted in on his territory were afraid of him.
Magic A Is Magic A: In the Harvey comics, at least, all ghosts have a very specific set of powers, and it is possible to make 'ghostproof' materials that they can't phase through.
Once per Episode: The Famous shorts became increasingly formulaic as time went on. Casper feels lonely, Casper tries to find a friend but everyone runs away from him, Casper finally finds someone who will overlook the fact that he's a ghost and be friends with him usually after he helps them get out of some kind of trouble. The friend is then menaced by a large baddie, at which point Caspar steps in and demands, "You leave my friend alone!" at which point the baddie runs off in fear. All ends well. Rinse and repeat. Famous Studios had a hard time coming up with original ideas for Popeye, too, incidentally. Lampshaded in an episode of Cheers:
Norm: I don't get it. Start of the cartoon, Casper has no friends. End of the cartoon, he has friends. Start of the next cartoon, he has no friends again. What happened?
Cliff: I think it's obvious what happened. Casper was quenching his thirst for blood.
There is also the phrase "A G-G-G-GHOST!" in just about every short.
Our Ghosts Are Different: Let's start with throwing the idea of ghosts being dead humans out the window (except for the movies).
Our Monsters Are Different: In Casper's Scare School, many of the creatures that would normally be former fleshies were never fleshies. Mantha was always a zombie, Ra was always a mummy, etc. The only characters that were confirmed to be formerly human were Fly Boy and his father.
Public Domain Soundtrack: In "The Friendly Ghost", when the other ghosts are in the sky and about to fly into the buildings, the music playing is an excerpt of Felix Mendelssohn's "Rondo Capricioso, op. 14".
Slave to PR: Ghosts are supposed to be scary, so Casper is feared by humans for being a ghost and looked down upon by other ghosts for not being a "proper" ghost. Other ghosts enjoy scaring, but also take it very seriously and do it because it's a ghost's purpose in death life. There are even scaring experts that the Ghostly Trio have either consulted to improve their scaring, or gone up against to prove that they were the best (in a Good Old Fisticuffs sorta way: the others and their fancy techniques are no match for tried-and-true invisible playing with objects or yelling of "Boo!"). One of the cartoons even featured a ghost school that taught haunting. Spooky protects his reputation as the number one scarer at any cost.
Strictly Formula: Nearly all of the cartoons are virtually interchangable from each other save for some superficial settings. One Famous Studios animator, Lee Mishkin, even had a quote about it:
"With the Casper series, you never knew what picture you were working on, because they were all exactly the same."
Terrible Trio: The Ghostly Trio, in every incarnation they've ever been in.
The Verse: Frequent Crossovers with other Harvey comics, even Richie Rich (though he's never sure he didn't dream his Casper encounters). Some arcs have brought together all Enchanted Forest characters.
Wild Take: The Casper series has some of the weirdest, sometimes outright grotesque, wild takes of the Golden Age cartoons, and the original shorts have them at least once per short. The over-the-top runaway gags or transformations after the takes (a whale scared by Casper runs off on its tailflukes, a zookeeper hides - then runs away - inside a lion, another group of zoo staffers combine together into a wheel when they see him, etc.) are the icing on the cake.
Halloween Town: Spooky Town, a city segregated into numerous boroughs for different supernatural beings that compete with each other to scare normal humans and ensure that the local Sealed Evil in a Can stays sealed.