Totally agree Liverpool. Not to far behind is City. And maybe in the future that will become more fierce. But for now and maybe forever, Liverpool.
Man U Rival

It's also worth bearing in mind that a huge number of people on manutd.com have no idea about our history, and probably couldn't find The European Capital of Trophies on a map.
There is still a big rivalry with Arsenal, but nothing like it was when Keano and Viera were playing.
Traditionally, Arsenal are below Liverpool, City, even Leeds, and possibly a few others. As TheCad said, these rivalries with Arsenal and Chelsea are relatively new, and would disappear more quickly if there were a couple of divisions separating the teams.
Liverpool on the other hand, could be playing in the Conference (oh we can dream), but the police would still mount a huge operation to cover what is one of the most volatile fixtures. It didn't get much coverage, but the Scouse B******s trashed the East stand when we DUMPED THEM OUT OF THE F.A. CUP the other week
Who do you think is Man U's biggest rival?