The Rebellious Spirit is an individual who has a beef with society. A large one. One large enough to make that person want to break all the rules, just because.
Rebellious Spirits go beyond the stereotypical "rebellious teen" and include people who flagrantly violate rules and social norms, act eccentric or weird, and often don't care what people think about them. Sometimes they even lack normal friends and usually lack good posture. They may be Chaotic Good, Chaotic Neutral, or Chaotic Evil, but they are always chaotic.
Note that this usually does not mean that the individual will lack their own rules; it's just that their rules tend to be different from those of everyone else in the given scenario. It will usually be everyone else's rules who this individual will be rebelling against, not their own. Someone who doesn't have rules at all, will probably be Chaotic Evil, or periodically do things purely For the Evulz in any case.
Sometimes, this person is the Anti-Hero or even The Unfettered. Can often prove to be a Combat Pragmatist when the fighting starts. May feature in an All Girls Want Bad Boys plot.
The eponymous Haruhi Suzumiya can be seen as a Rebellious Spirit, being eccentric, egocentric and borderline sociopathic as she is, she screws the rules and social norms on an everyday basis. If it's because she hates rules, hates those rules or just doesn't care about them is arguable. Please note that we might not be talking about laws and regulations of a society, we might be talking about laws of nature.
Naruto Uzumaki hated the fact that his village ignored him. So what does he do? Defy all the rules and uses his ninja skills to play pranks on unsuspecting villagers/ninja(including the Hokage), his worst prank being when he defiled the somewhat sacred Hokage monument. Of course he got more heroic as the series went on, but he still has a problem with authority, preferring to do things his own way instead of listening to his superiors. Considering his early experiences with adults, he can't really be blamed.
Azaka Kokutou from Kara no Kyoukai is this explicitly (highlight of this is her unabashedly declaring her love for her brother and moving away to be adopted by her uncle and to enroll in a boarding school specifically so Mikiya will stop seeing her as his little sister), thanks in no small part to her origin being taboo.
Eren Yeager from Attack on Titan, a rebellious youth that spent his childhood getting into fights and accusing others of being livestock for accepting life within the Walls. He turned out to be Properly Paranoid, but remains openly critical of others that don't want to fight the Titans head-on, and his career within the military is filled with him calling people cowards and challenging every social norm.
SpiderMan: Was generally a rogue free spirit hero who defied the local authorities constantly and performed acts of vigilantism. He also generally scoffed at help and was a pure solo act hero contrasted to the Avengers. Nowadays, he's more mature and dedicated to working together with people and being more considerate of the law.
Lawn Dogs has two of these as its main characters. Adult Trent and kid Devon both have no friends their age. Trent does crazy things like holding up traffic just so he can skinny-dip off of a bridge. Devon is even crazier. She violates rules frequently, such as leaving town when told specifically not to, and even breaks the law and steals chickens for fun. She has many weird eccentricities as well.
Literary / Film / Truth in Television example: Fear and Loathing in Las Vegas features two characters (Raoul Duke and his attorney Doctor Gonzo) who run a two-week, Sixties-style drug binge through the streets, hotels and convention centers of early-SeventiesLas Vegas, with no real motivation beyond seeing how much they can get away with. And they get away with a lot: showing up stoned at an anti-drug convention, chasing motorists down the Strip, trashing two hotel rooms, racking up several hundred dollars in room-service charges and never paying... Truth in Television part comes in when you realize that this novel is based on two real-life wild weekends spent in Vegas by Hunter Thompson and Oscar Acosta.
Thompson also viciously pranked friends and strangers alike, set out to spraypaint "Fuck the Pope" on a yacht due to compete in the America's Cup the next day, chronically missed speaking engagements just because they were scheduled, and, decades before Sascha Baron Cohen, constantly reported people instantly willing to believe vicious and absurd rumours he'd just invented.
He's Crazy Awesome because of this behavior, although you may not like the overall lack of ethics.
Charlie of Dead Poets Society drifts into this sometimes; it's not as clear at the beginning, but by the time he nearly gets himself expelled for a prank he is showing more traits of this. When he actually does get expelled, though, it's a much more principled matter.
The Steel General, from Roger Zelazny's Creatures of Light and Darkness, is an immortal cyborg who has dedicated himself to supporting insurrections across the universe, coming to be known as "The Prince of Revolution". He plays a mean banjo, too.
Jack Random of the Deathstalker series is literally described as a "professional rebel." The same establishment has been his enemy through all his insurrections, but he takes it as his entire role in life.
Eventually he successfully brought down his enemy and became a key leader in society's reformation... only to discover that the changes were too limited for his tastes. So he created an insurrection against the government he just helped found. His reasoning was that there should always be somebody to oppose the government lest it fall to evil.
In Discworld, the entire race of gnomes is reportedly made of this trope, happily trouncing all over even those rules which go without saying in normal society, like "Do not attempt to eat this giraffe".
Despite being Lawful Good in the extreme, Sam Vimes shows tendencies in this direction as well, even after becoming a Duke. This is probably why he employs at least one gnome in the Watch.
The Stainless Steel Rat: an almost childishly rebellious hyperactive action hero. Slippery Jim is at war with normal society as much as the villains, and his greatest weapon is his willingness to transgress the bounds, rules and world-view of other people. That said, he's also The Fettered, being an Actual Pacifist.
The Doctor from Doctor Who was this on his home planet. Partly because he found Time Lord social norms unbearably boring, and partly because he was disgusted and disturbed by their practices and elitism, he stole an antique time machine from a museum and ran away to explore the universe as an act of rebellion, since Time Lords were masters of time travel yet refused to ever use their powers and preached non-interference.
The Minmatar in EVE Online are characterised as loving freedom so much after having been enslaved by the Amarr for a thousand years that a large number of them can't even live comfortably with the government they themselves formed.
The Orks of Warhammer 40,000 invert this: "Yoofs" who have difficulties fitting in with an Always Chaotic Evil society may get tired of being told to do whatever they feel like. Some of these malcontents run off to join the Stormboyz, Orks obsessed with military discipline, marching, and uniforms. Since they make good Jet Pack-equipped assault infantry, the rest of the Orks put up with their deviant behavior.
In "Defying Gravity" in Wicked, after learning the truth about the Wizard of Oz, Elphaba (the Wicked Witch of the West) declares "I'm through accepting limits 'cause someone says they're so / Some things I cannot change, but till I try I'll never know!" and begins her "campaign of terror."
Power Gig: Rise of the SixString has the Rise Clan. They will not put up with anyone who exerts any sort of control over them, legal or otherwise, and do not care who knows it. They don't like the Followers of Zhen, whom they feel are chained to a useless restrictive dogma, and see The Riffriders clan as a group that is all talk and spends too much time having fun and not enough actually rebelling against things. The only reason other Rise members can put up with each other is that they all agree that control sucks and don't care what other Rise members, or anyone else for that matter, thinks of them.
Grand Theft Auto IV: The Lost and Damned has Johnny Klebitz of the eponymous Lost Motorcycle Club. Almost everyone he meets treats his attempt at sticking it to the man as unimportant, which irritates him more than being treated as a dangerous outlaw would.
In the episode "Mona Leaves-A," Lisa Simpson of The Simpsons inherits her late grandmother's "rebellious spirit." This later turns out to be part of Mona's posthumousplan as she and the rest of her family use their inheritance to help Homer escape from Mr. Burns.
In "Over Bearing" on Care Bears: Welcome to Care-a-Lot, prim and proper Peter becomes this after Funshine Bear and Grumpy Bear start encouraging him to break rules. "I want to break all the rules in Care-a-Lot! I'll jump in the leaves! I'll yell in the house! I'll never follow another rule again! I'm a bear in the woods and I listen to no man! Mine!"
In Hercules the animated series, Electra and her associates are this, with Electra far beyond the others—we don't know why they're against Greek heroes or "the establishment" other than that they seem to be a Take That at Goths and Hipsters. Electra is even worse since she can summon the Furies when she's upset, usually when someone disagrees with her views.