If any cartoon was defined by a rivalry, it was Tom and Jerry. While Tom is usually the aggressor, he seems far more obsessed with winning than hurting Jerry, while Jerry seems far more content with humiliating and outwitting his foe.
Take a Red Oni, Blue Oni pairing and make them rivals, and you've got Bugs Bunny and Elmer Fudd's conflict down perfectly. The yearn - possibly obsession - the serious-minded Elmer has over nabbing the free-spirited Bugs has spanned over seven decades and is still going strong. And it seems in all that time, Elmer never learns to see through Bugs' lame disguises.
Daffy Duck is a rival to Bugs in a different way. An egotist to the core, Bugs is the Always Someone Better whom Daffy can never seem to best.
Sylvester to Tweety. At first, it seems like a predator-prey thing, but like the case with Tom and Jerry, this has become a personal obsession for Sylvester. Why else would he be so dead-set on catching one little bird? (In fact, one cartoon ends with him deciding to fold and go to a restaurant for spaghetti, but this attempt to quit didn't last.)
Dear lord, the Roadrunner to Wild E. Coyote. No matter how many times his ill-conceived plans to catch the speedy bird fail, often with himself getting hurt in the process, Wile E is certain to be back on his feet soon with a new Acme product and a renewed drive to do so. Sometimes a viewer can't help but admire such determination.
In DuckTales, Scrooge's chief rival is the unscrupulous Flintheart Glomgold. While Scrooge's avarice is often tempered by compassion and charity, Glomgold has neither, portrayed as an unrepentant miser, tycoon, plutocrat, and general antagonist.
Craig Tucker and his friends are this to the main four boys in South Park
Mandark is Dexter's rival. The two clash a lot though Mandark does have a serious thing for Dexter's sister Dee Dee.
Plus, the episode "Dee Dee's Rival" did a Gender Flip of this trope by pitting Dee Dee against Mandark's sister, who was apparently created specifically for the episode and proceeded to never be mentioned again. She nonetheless went through the complete arc of this trope during her brief screen time, right down to the grudging respect (by the end of the episode, she and Dee Dee have decided to be friends).
Squilliam has been Squidward's rival since highschool band class.
Bubble Bass was Spongebob's rival, but was phased out after season 1.
Mr. Krabs and Plankton are fierce business rivals. They were once friends until they had a falling out over the Krabby Patty recipe, which Plankton will now stop at nothing to get. Of course, while It's Personal for Plankton, Krabs is Only in It for the Money (which is, of course, the only reason he has for anything).
Endive has been Mung Daal's rival in the catering business for years. Panini should be Chowder's rival, but she chooses to be his YandereStalker With a Crush instead, so Chowder's actual rival is Gorgonzola.
Apparently Panini was originally supposed to be just that when the creator pitched the series. He decided to retool her when it got picked up.
On American Dad!, Roger invokes this trope in a horse race when he tells one of the other jockeys he got it on with his wife. Roger did this deliberately because he thought the race would be more interested if he had a rival to compete against.
When Steve put together peewee football team to compete against his father's one of his recruits is a Chinese kid named Li-Fung who doesn't speak English. Steve's Japanese friend Toshi attacks for stealing his bit.
Cindy Vortex towards Jimmy, NON-STOP. She does however get a Relationship Upgrade from his rival and enemy in the first season/movie to his rival-friend in the second season to his girlfriend in the third season.
He later gained another rival in rich kid, Eustace Strych, possibly to give Jimmy someone else to compete against after he got a Relationship Upgrade with Cindy.
In the episode "Lemon of Troy", Bart has Shelby (a doppelganger from a rival town, Shelbyville) as a rival. In fact, Milhouse also has a rivalry with his own counterpart (also named Milhouse) from the same town.
In "Lisa's Rival", Lisa forms a one-sided rivalry with the new girl, Allison Taylor. Too bad she is Demoted to Extra for the remainder of the series, even after their reconciliation. (It seems any new girl in school is destined to be Lisa's enemy - see also "Lard Of The Dance" and "Bye Bye Nerdie.")
In can also be said that Marge has an occasional rivalry with her gossiping fair-weathered acquaintance, Helen Lovejoy.
Mr. Burns also has his own business rival at the Shelbyville Nuclear Power Plant, Aristotle Amadopolis.
Then there's Seymour Skinner's rivalry with Principal Valiant of Shelbyville Elementary School, due to the school's better funding and academics.
Lisa claims that Maggie's arch-rival is Gerald, the baby with one eyebrow, though we never see them do anything other than glare at each other while ominous music plays.
And then there's Homer's iconic and extremely one-sided rivalry with his neighbor Ned Flanders. Homer despises him for being successful in all the ways that he isn't, but Ned views him as a close friend.
Posthumously, the town founder Jebediah Springfield had one, his old partner Shelbyville Manhattan. An argument over giving men the right to marry their cousins started the ongoing rivalry between Springfield and Shelbyville.
Mr. Wilter and Reggie Bullnerd from ChalkZone. Mr. Wilter hates creativity and shuts down Rudy's art at the first chance he gets. Reggie, meanwhile, is just a dumb bully who has it out for Rudy, leading to the creation of Bully Nerd, a hulking chalk drawing of Reggie that chases Rudy and Snap for an episode.
Trixie is often portrayed as Twilight Sparkle's rival in fan works, and the episode "Magic Duel" canonizes it. (Twilight tends to think differently.)
Rainbow Dash finds a rival in Lightning Dust "Wonderbolts Academy". Unlike Trixie, Lightning is clearly able to match Rainbow in ability, the first rival to actually achieve this with a mane character.
The Flim-Flam Bros. establish themselves as rival apple cider sellers to the Apple Family in "The Super Speedy Cider Squeezy 6000".
Pinkie Pie finds a rival party thrower in Cheese Sandwich in "Pinkie Pride", who initially shows her up as a party planner. Unlike previous rivalries, he is firmly a Friendly Rival by the end.
In "Rarity Takes Manehattan", Suri Polomare becomes Rarity's rival as a fashionista, and like Trixie, does not match up to Rarity in ability and relies on being an Opportunistic Bitch to get ahead.
The Misfits are the rivals to Jem And The Holograms in Jem. They're both extremely popular, 1980s American pop bands who are in the same area. The Misfits frequently try to sabotage Jem And The Holograms but they always fail.
In the first season of Winx Club, Riven is the Rival to Sky's hero. Although he acts antagonistically to everyone, Sky is the only one who does anything other than get annoyed or brush it off, and there are resultant duels. His behavior is genuine but gets Turned Up to Eleven when Darcy takes control of him with magic. After that stops, Riven joins the Band of Brothers and becomes a Friendly Rival.