What are some famous arch-rivals?

That reminds me, do sports rivalries count? If so, how about:

–the New York Yankees and the Boston Red Sox

–(until 1958) the New York Yankees and the Brooklyn Dodgers

–the Dallas Cowboys and the Washington Redskins

–Muhammad Ali and Joe Frazier

Hawkeye Pierce and Charles Emerson Winchester.

Cardinals and Cubs.
Dodger and Giants, which rivalry followed both teams from NY to CA.

Except that they got along swimmingly. Mycroft was smarter, Sherlock knew it and admitted it gracefully, and neither one cared, because they weren’t interested in doing the same kind of thing. Mycroft was the sit back, plan, and send someone else to do the dirty work. Sherlock enjoyed the dirty work; it let me be theatrical; he’d have gone crazy–well, crazier–cooped off like his older brother all the time.

Having just started watching The Adventures of Brisco County, Jr. on DVD…

Lord Bowler is Brisco’s arch-rival (John Bly is his arch-nemesis).

Elaine Benise and Sue Ellen Mishke

Snoopy and the Red Baron

Jimmy Kimmel and Matt Damon

Perry Mason & Hamilton Burger

J.D. and the Janitor

Well, the Saint had Claud Eustace Teal;

And they both battled the ungodly and brought, or attempted to bring, justice to the masses.

House & Tritter

Would a relationship based on one-sided hostility, like Salieri and Mozart in Amadeus, count as arch-rivalry?

I would say House and Cuddy. Though they’re not bitter enemies, they’re constantly sparring, each one trying to get the upper hand. And House never once, as far as I could see, had a shred of respect or admiration for Tritter.

Luke & Darth Vader?

Hunter Thompson & Richard Nixon? (yes, they were in RL, but they expressed themselves through fiction)

Wonder Wart-Hog & The Masked Meanie?

Cole Richards and Max Power.

Randall McMurtry and Nurse Ratched.

Dorothy and Miss Gulch / the Wicked Witch of the West.

Flash Gordon and Ming the Merciless.

the Doctor(s) and the Master(s).

Karen Walker and Beverly Leslee.

Saffy Monsoon and Patsy Stone.

Betty and Veronica.

Homer Simpson and Ned Flanders (at least, according to Homer).

Maggie Simpson and the One-eyebrowed baby.

God and the Devil.

Ranma Saotome and Ryouga Hibiki in Ranma 1/2.

OK, maybe not famous…