Blue always frustrated me being one step ahead of me and taunting me all the way to the Elite 4. Beating him and becoming Champion was easily the most satisfying battle I've had in a Pokemon game.
Silver was a nasty guy. He stole his starter! It was interesting watching him slowly becoming a decent person. I didn't get that rivalry feeling from him but he had good character development. I appreciated that.
Everyone else is a friendly rival and I think that's something that should be developed over time. You should start out in conflict with your rival and over the course of the game gain a mutual respect for each other or perhaps even become friends. Starting out as pals takes the fire out of the battles for me.
3DS Friend Code: 2466-2280-9220 ACNL Town: Cascadia "Way past cool!" - Sonic the Hedgehog