Of all of the possible rivals, it's probably going to come down to Paul and Gary. Barry, Trip, Morrison, and any other person who could fit the description of rival really can't compete.
Between those two, my vote would be for Paul. Sure, I didn't like Gary. He had an inflated ego and he was irritating, but at least he cared for his pokemon. If I met someone like Gary, I'd say "What a jerk" when he left the room. Paul was smarter and a better trainer than Ash, which would make for a good rival already, but worst of all he was doing better by doing things that are against Ash's morality and core beliefs.
To me, it's a difference of a rivalry of ambition between Ash and Gary, and a rivalry of ideology between Ash and Paul. The latter just seems more important.