Spike to Angel, as revenge for all those times Angelus humiliated him and stole his girlfriend. He also butted heads with Riley on frequent occasions.
Reaches a new level in Angel, due to the Shanshu prophecy involving a vampire with a soul.
Angel unwittingly provokes Lindsay's wrath at every turn, from ruining his court cases, to stealing his would-be girlfriend (Darla), to having the entire firm handed to him on a silver platter. ..And then Angel has sex with Lindsey's new girlfriend, Eve, under the influence of spell — and oblivious to Lindsey's connection to her. When Lindsey consoles himself that Eve is one of the only things in his life that Angel "never got his mitts on", Eve wisely holds her tongue. A good thing, too, because otherwise his brain might have imploded.
Sharon Raydor shows up as this to Brenda in Season 5 of The Closer, possibly because Commander Taylor is now something of a Lancer. After working together (quite well) to solve a case in the fifth season finale, they agree that they "simply don't like each other." This turns out to be a big fat lie. Raydor went on to take the helm of the Major Case Squad in After ShowMajor Crimes following Brenda's departure from the LAPD, and has proceeded to kickass.
Degrassi and Lakehurst were rival schools, which caused problems when Lakehurst burns down and the students have to move to Degrassi. One main character even gets stabbed to death by a Lakehurst student. Liberty had a rivalry with Damian, who was to be Lakehurst's valedictorian and became her competition for the position when he became a student at Degrassi.
Grey's Anatomy: Seattle Grace and Mercy West hospitals. Similar to the Degrassi example, they merged when Mercy West closed down. Some Mercy West characters were killed off, though the others have been accepted as part of Seattle Grace.
Many of the secondary Heisei in Kamen Rider play this role at first, then eventually become an ally or a true foe.
A considerably more benign version of this trope is Prince Arthur and Lancelot from Merlin. Both of them respect the other, but both are painfully aware of their mutual feelings for Guinevere.
For each series Arino does multiple games on. Most are simply the Big Bad of multiple games, such as Bowser, but some are enemies he has a lot of trouble with, most notably Red Arimer/Red Areemer/Firebrand and the birds.
Turk had a rival in Bonnie during the early seasons of Scrubs.
Bimpe from Tinsel is Telema Duke's rival, both in the industry and for Soji. Kwame is Soji's rival. Brenda Mensah is Fred Ade-Williams's rival.
Lex from The Tribe is a rival to Bray, as well as most other males on the show with any authority. He's also a thug and a petty criminal who thinks he's much more of a Magnificent Bastard than he really is.