Manchester City fans, as a rule, detest Manchester United. The feeling may not be reciprocated, as United supporters may see Liverpool or Arsenal as their archrivals(due to City's not winning anything for 42 years).
Ajax Amsterdam and Feyenoord in the Dutch football league.
Greek football has Panathinaikos-Olympiacos Piraeus. You know this rivalry qualifies when it is known as "The Derby of the Eternal Enemies."
Germany has Borussia Dortmund vs FC Schalke 04, known as the Revierderby (Ruhr Derby).
Italy has Lazio-Roma and AC-Inter.
Portuguese football has a three-way arch-rivalry, with Porto, Benfica and Sporting Lisbon fans all detesting each other.
Spanish football has many rivalries, but none come close to the Real Madrid-Barcelona rivalry known as el clasico.
Turkish Football has Galatasaray and Fenerbahce, and to a lesser extent, Besiktas.
In Brazil, not only inter-city teams are arch enemies, but some manage to cross state borders as well. A lot of contested important games between Atlético-MG and Flamengo (the country's most popular team) made the former hate the latter as much as city rival Cruzeiro.
The Los Angeles Dodgers and the San Francisco Giants have been arch rivals since the 1800s, from their days in New York, all the way across the coast when they both moved to California.
The New York Yankees and the Boston Red Sox, the American League's, and possibly Major League Baseball's, bitterest rivalry.
Completing the trinity of baseball rivalries is Chicago Cubs vs. St. Louis Cardinals. This due to the long history, close proximity and the fact that Cardinals are one of the most successful teams in history and the Cubs... are not.
In basketball, there really is no better rivalry than that between the Lakers and the Celtics. This one is born out of success, as opposed to any local enmity.
The Miami Heat, the Chicago Bulls and the New York Knicks have a three-way rivalry.
The New York Liberty and the Detroit Shock were very intense WNBA rivals, until the latter was sold and moved to Tulsa.
The Detroit Pistons and the Los Angeles Lakers, Bill Laimbeer and Rick Mahorn in particular. When they were reunited as WNBA coaches on opposite sides, their egos got in the way, and both teams got into a fight.
The Montreal Canadiens have not one, but two: the Boston Bruins and the Toronto Maple Leafs. For a while, the Habs had three archenemies. In addition to their enmity with the Bruins and the Leafs, the Canadiens also had an ugly relationship with the Quebec Nordiques, who played in the provincial capital of Quebec City. This seems to have died down after the Nordiques moved to Colorado and became the Avalanche, though.
The Leafs have another rival similar to Habs-Nordiques, the Ottawa Senators.
NCAA hockey brings us the Battle of Commonwealth Avenue: Boston University vs Boston College, which stretches all the way back to BU's first hockey game ever on February 6, 1918. It doesn't help that the two schools are within 5 miles of each other and only a couple stops away on the Green Line.
Any Chicago and Detroit teams - Blackhawks fans in particular shout "Detroit sucks!" in every game, not only those against the Red Wings.
New York teams in all sports have fierce rivalries with teams from Boston and Philadelphia.
Michigan State University's football and basketball teams, both called the Spartans, are each this to their University of Michigan counterparts, both called the Wolverines, and the feeling is mutual.
Duke vs. UNC. One of the biggest rivalry in college sports (especially men's basketball).