What the title says, pretty much. Try to complete one of the Action Stages, or if you're mad enough, a whole story with only one hand on the controller. It's actually more doable than you think; recently I was playing this game and noticed that you could complete Sonic's Emerald Coast without even jumping or spin dashing. I really only tried Sonic's stages before writing this, but you can also try the other characters as well.
I'd recommend using your left hand for this challenge for easier access to the much-used analogue stick, as well as doing the Spin Dash Jump (spin dashing and jumping immediately after) to get to farther platforms, and the Homing Attack's dash to get a slight bit of momentum for short jumps.
It goes without saying that this makes the game considerably harder, especially the more platform-oriented levels like Red Mountain, Lost World and *shudder* Sky Deck. Do the best you can and compete for the best times.
And that's about it. Have fun!