Rivalry

The least important are the more fun...

Entreprise/Corporate:
Boeing - Airbus
Coca-cola - Pepsi
Microsoft - Macintosh (Apple)
Hewlett Packard - Texas Instruments
NASA - Russian Space Agency (RKA)
Nokia - Motorola
Yahoo - Google
Nike - Reebok
ATI - nVidia

Geographical & Political:
Pakistan - India
Israel - Palestine
Argentina - Brazil
Ireland - United Kingdom
China - Japan
Iraq - Iran
Iraq - USA
USA - Canada
England - France
Germany - England
Roman Empire - Egyptian Empire
Portugal - Spain (XI ~ XVI centuries)

Sports:
Mohammed Ali - Joe Frazier
Bjorn Borg - John McEnroe
River Plate - Boca Juniors
Boston Celtics - LA Lakers (80s)
Mike Tyson - Evander Hollyfield
Liverpool - Manchester Utd
AC Milan - Internazionale
Fenerbahçe - Galatasaray
Real Madrid - Barcelona
Flamengo - Fluminense
AS Roma - Lazio
Sporting - Benfica
Carl Lewis - Ben Johnson
Pete Sampras - Andre Agassi
NY Yankees - Boston Red Sox
Detroit Pistons - Chicago Bulls
Australia - New Zealand (rugby)
Chicago Bulls - NY Knicks (90s)
Nacional Montevideo - Peñarol
Steffi Graf - Monica Seles
Glasgow Rangers - Celtic
Benfica - Porto
The Rock - Hulk Hogan
Argentina - Brazil (football)
Tonya Harding - Nancy Kerrigan
Pakistan - India (cricket)
England - Australia (ckicket)
Dallas Cowboys - Washington Redskins

Other:
Tom & Jerry
Elmer Fudd - Bugs Bunny
Wyle E. Coyote - Roadrunner
Austin Powers - Dr. Evil
Batman - Joker, Penguin, ...
Superman - Lex Luthor
Cats and dogs

I like all type of rivalries and always cheer for one side, even if it's Turkmenistan vs Kazakhistan, men vs women, north vs south, socialists vs liberals or anything else. But the main choices are... Benfica, Chicago Bulls, Argentina.