Giving it to the police, you assume they'll have some sort of moral decency, and influence, but probably not. I mean, they probably wouldn't know what to do with it, and you've already moved the private key at that point.
It depends on the country, for example, I wouldn't entrust this to police forces of a country that has high corruption index. I don't even think it's wiser to entrust this job to any policeman or policewoman because even they can cheat in this case without punishment, it's not a bank account after all.
It's very easy for someone to go Ah, I've had this for a year now, and no one has claimed therefore it's mine, but I don't really agree with that either. So, I'll agree it probably isn't a accurate statement, since I personally wouldn't be comfortable with moving it at all, but then am I prepared to allow someone else to steal it when I could've prevented that? Tough decisions.
This is a massively complex problem, which doesn't have an easy answer that it might appear to have.
It's not that massively complex problem. Did you post on Reddit? Did you post on bitcointalk? Let's say yes. Did anyone came up with real proof that it belongs to them? There are a lot of ways to prove if you really want to and it's not only signing a message. If no one came and you waited for weeks, months, then what's wrong if you consider that from now on, it belongs to you? You haven't stolen from someone. Someone lost it, you tried your best to find the owner but owner did least to find you, then it belongs to you. You push yourself too much, you think like you are the killer if you tried to assist wounded person but failed to escape him.