The first place is surely taken by Barcelona, but Atletico Madrid, the city rival, is not far behind. Other rivalries include Valencia, and most certainly Athletic Bilbao. Although the rivalry risen against Valencia and Atl. Madrid are mainly exist on a football level, the same cannot be said for the other two. Athletico Bilbao are the biggest football team of the Vasque region of Spain, which want's to make an autonomous county, so the rivalry has a political level to it. The same more or less goes for Barcelona, with the roots of the rivalry going back to the days of Franco.
Lamborghini is the main rival for Ferrari.
Their main rival is Manchester United
The name of Edison's main rival was Nikola Tesla.
The main rival of FC Schalke is Borussia Dortmund.
There are two main teams in Turin. They are Juventus and Torino.
England is Australia's main rival at the World Championships and Commonwealth Games.
Chargers
The main rival now is Alabama. Following that would be Auburn.
Spain was their main adversary
Leon Trotsky was Stalin's main rival inside the Soviet Union. Stalin had him deported and finally assasinated. Adolf Hitler was Stalin's main rival outside the USSR. The Russian army invaded Germany and found Hitler dead.
The main rival is the D.C. United, but New England Revolution and the Philadelphia Union are also rivals.
The main rival of the BYU Cougars is the Utah Utes. Their annual football game is known as "The Holy War"