Stewart Copeland was one of the three main rock influences when I first started playing (Steve Smith and Jeff Porcaro being the other two).
His style is so unique yet pretty subtle. You don't see him drawing attention to himself. You find yourself listening closely to his hats and other intricate parts and thinking...wow...that is so much harder than it sounds.
This is my favorite bit of his that is a bit more showy than his normal stuff. This was the first drum part I ever broke down piece by piece to learn:
I find that I use that first break probably 3 or 4 times a night per gig! I almost forgot where I even picked it up, until No Time This Time came on my mp3 player for the first time in forever.