I'm probably being biased but based on trouble on and off the pitch in recent years the biggest rivalry in english football has been Villa Blues (Aston Villa and Birmingham City). Just because we're not competing for trophies it gets overlooked but both sets of supporters and even some of the players (Mellberg) really hated each other.
Other than that, in Britain Celtic Rangers is still pretty intense considering they've been playing each other 4-5 times a year for god knows how long.
Also Tottenham Vs Arsenal.
But as for Villa Vs Blues we all know who is obviously way better, Aston Villa.