DAI: LS toggle...character only moves at a slow jog

I know on PC there are two speeds for mounts, one a slow jog, and then a sprint button to pick up the pace. 

As far as Multi-player goes, I don't play it but I have found that running with your weapons out slows you down, as well as certain skills being active.  In single player, I hit Disenguage to sheath weapons and run faster when I need to in combat.