I like the idea of contacting the cops and seeing if you can locate the owner.

You can investigate more with the local homeless folk and maybe get a name. Then this big adventure could ensue which involves you visiting the guy in jail in an orange jumper, maybe a glass divider between the two of you, all that good movie scene stuff. You could be his only visitor after all. Milk it like you have something really important to say as his one visitor you've been trying to track down for so long, and your his only contact with the outside world, then say

"Soooo yeah...your bike is pretty cool. Can I have it?"