It would work the same on a Stage as it would on a carpet.
It isn't microfiber, obviously, so the iRide doesn't have the no-slip principle, but it should stay put as long as your weight is on top of it. However, if you own the Stage, make sure you can straddle the Stage first as the Stage can be pretty wide to straddle.
As for on the Wedge, it's an inclined surface, so the iRide doesn't have any no-slip properties to it, so I don't imagine it'd stay on very well. It depends on how you were laying and how much of your weight you were putting on the iRide. I'm trying to figure out a position as to why you'd use the iRide with the Wedge, but I can't. x.x Maybe if you give me an idea of what position you have in mind, I can give you a better answer of whether it will work. (We own both)