Wow, my top pick showed up real quick on the first page. Big Boss in MGSV has an awesome running animation. It just works when you're bolting around in a tense situation. It also flows perfectly into dives.
Amaterasu's run in Okami is also good. The trail of grass and flowers she leaves behind, how she stretches out wile zipping around (it's a joy just darting around with her at top speed), the speed increase once you've been clopping around enough.
Kirby in Kirby's Epic Yarn and his little yarn car animation. One of the cutest things in gaming, complete with "BEEP BEEP" and squish animation when he hits something at speed.
Here's one no one has mentioned yet: The Arisen in Dragon's Dogma: Dark Arisen. When you book it in that game, you BOOK IT. When you change direction the character experiences a shift in weight and slides a bit before turning around. It's just as intense and ferocious as the encounters you find yourself in in that game.