Biggest rivalry in football (that’s pansy soccer to you Jamie
) is without a doubt Glasgow Rangers vs. Glasgow Celtic (they are know as the Old Firm) basically you don't want to wear the wrong colour shirt in some of the schemes of Glasgow - generally there's about 5-8 stabbings after every derby (bloody glad the scum aren’t allowed guns here) and its a f*cking nightmare going out on a Saturday night in Glasgow city centre after one of them (folk are drinking all day and there’s a fight in nearly every pub).
The rivalry really comes from the sectarianism that is rife in West Coast Scotland and Northern Ireland. Rangers are traditionally a Protestant Loyalist (i.e. pro United Kingdom) supported club and Celtic has a Catholic pro united Ireland roots. What this means is the Troubles in Northern Ireland spill over to the football chants and regalia. There used to be regular collections for the Irish terrorist groups (both sides) from fans in the 70's - 80's.
Just to brief you on some of the shocking things associated with the clubs in the past - Rangers refused to sign a Catholic player for many years (i.e. 100ish) and the first one the signed (in the late 80's) was a former Celtic player (big trouble - some fans refused to count the goals he scored for the club all season). A Northern Irish Catholic Celtic player (Neil Lennon) had so many death treats he had to retire from playing for his country. Numerous players from both clubs have been pictured holding flags/scarfs with terrorist logos (Celtic fans notoriously fly PLO flags to show the link between the IRA and the PLO). Recently one Rangers Player (Bob Malcolm) autographed his name to a school kid fan with FTP after it (F*ck The Pope).
Although I live nearish here and lived in Glasgow when at Uni, I refuse to support either of these bigoted scum (Motherwell FC for me!), but bear in mind these are by far the biggest supported clubs in Scotland and are among the most supported clubs in the world with large ex pat supporters groups in the US.
There are other big rivalries in British football (i.e. Arsneal vsTottenham, Liverpool vs Everton, Hearts vs Hibs) but not to this scope. Although I will say that some of the Italian clubs are nuts and just last week a fan brought a bomb to the stadium which has unfortunately killed a Policeman leading to the suspension of all football in Italy for the time being.