Now what do I do with this "stolen" bike?

So a school friend of mine couldn't get his bike free of the lock two weeks into last fall semester. I spent the whole semester trying to get him to come with me, because I was sure I could get it for him. He went once, and I couldn't do it then, and he never let me try again with him around. The day before campus safety came to donate all the abandoned bikes, I got it free. Campus safety saw me doing it the whole time, but didn't say anything. I posted a picture of it in my room, and my friend said he'd pick it up, and we set three times, but he was always "too busy," or "too tired." I'll bet he doesn't even really care.

I don't know how to ride the thing, and it's all beat up, but I'd like to learn. Should I offer to buy it? Should I get it appraised first? How can I prove that I actually did buy it if I end up buying it off him?

It is a bicycle, not a motorcycle.

Doesn't sound like he wants it to me. Depending on the laws in your neck of the woods, it could be considered property abandonment. You don't want to go the legal route on your friend though. Just say "hey, we set up 3 different times for you to come pick this thing up and you haven't been able to make it, I really want to learn how to ride and if it sits here any much longer I'm going to assume it's ok for me to teach myself on it" He's either going to come and get it or not. Either way, you get what you want. You shouldn't have to buy it though, unless he's going to pay you for your months of storage.

A motorcycle? Or a bicycle? If it's a bicycle you don't need to prove anything. Ride it if you like it, or sell it, or give it to a friend.

It sounds to me like he doesn't want it, so as long as he knows you have it, just ride it till he says something. It would be wrong not to let him know you have it, because he's your friend.