prokopetz:

Sonic: Would find a way to safely channel the Ring’s power in spite of the metaphysical impossibility, use it to power up and fight a heretofore-unhinted-at three hundred foot MegaSauron, then fumble it into the lava of Mount Doom in a moment of tonally incongruous slapstick.

Tails: Would quickly figure out how to dispose of the Ring, but be prevented from telling anyone by a series of increasingly unlikely interruptions, including a kidnapping by a gang of baddies who intend to blackmail his companions into revealing the location of the Ring, unaware that Tails himself is carrying it.

Knuckles: Would be tempted by the Ring, but lose possession of it before being called upon to make any moral decisions, because the inability to keep hold of dangerous magical artifacts for more than ten minutes at a stretch is the literal basis of his character.

Robotnik: Would seize the Ring eagerly, but entirely misunderstand its purpose and potential, and end up using it to power, like, a giant murderous toaster or something.

Shadow: Would cross paths with the Ring early on, but rapidly become side-tracked by a journey of self-actualisation only tangentially related to the actual plot. Probably fistfights Treebeard for some inexplicable reason.