The only rivals I'm really familiar with (in terms of exposure and interaction with Ash) are Gary and Paul (and Trip to some extent). Paul seemed to push the boundaries of what a rival does (like releasing Pokemon, dismissing "unfit" Pokemon, reluctance to team up with Ash during a battle IIRC, etc.), but Gary is classic. Ash and Gary hailed from the same starting point of Pallet Town, and flourished in each of their paths of being a Pokemon Trainer. Their climactic Silver Conference encounter and battle were outstanding (memorable for me, even), and both of them have matured in their own ways, taking cues from each other, at times.
Plus, Gary did seem to drive Ash to do better frequently, even if he seemed like a jerk while doing it. "Gary was here! Ash is a loser!" comes to mind. While Ash usually just got riled up earlier in the series, it seems that, later on, he uses Gary's taunts as fuel to the fire of Pokemon Trainer excellence. Finally, Ash and Gary seem to acknowledge each other skill, yet they still wish to surpass each other (mostly Ash nowadays, since Gary doesn't journey as much as he used to in terms of Training).
I haven't seem much of Trip, but despite his off-putting personality, he has potential, seeing as he wiped the floor with Ash (a "veteran" of at least Frontier Brain caliber, so to speak) as a rookie Pokemon Trainer.