Riding a bicycle is illegal on some roads, especially the highway (unless it's the only reasonable road, such as though a canyon).

That being said, far too many assume it's illegal to ride on the road, and "share the road" signs don't seem to effectively correct that misconception. I've even heard of drivers telling cyclists to ride on the sidewalk when they're already riding in a bike lane...

That being said, and as others have stated, riding on the sidewalk is not illegal in many areas, though it's less safe in most.