His body and everything else around him can handle it, so why not a bike? Like at a certain point, he'd be accelerating fast enough to just step out of his shoes and costume from his leg movement. Though he's often shown going from a standstill to moving faster than your eyes can follow, which should probably leave his clothes behind.

Probably bits of his body, too. Like the iterarions of him that could hit or exceed the speed of light are moving fast enough that running into air would probably feel more like a cement wall, plus a bunch of ionization effects just from the sheer friction.

There's that ask xykd about a baseball being thrown at some high portion of c and iirc, the conclusion was that everyone in the stadium and in some radius of it would die and that was just for a baseball. A full person would be an order of magnitude bigger. Even The Boys didn't quite get it right in their premiere, it would have been more violent than just his gf getting vapourized (though not sure that guy was going close to c, so it might not be "everyone within 100m of him dies when he runs", but he wouldn't need to run into someone to affect them at "just" faster than you can follow with your eyes).