Hm, I'd probably keep it and use it for personal gain, but that depends on the circumstance. If it could somehow be proven to me that this Death Note falling into the hands of humans was just a one time event, and by destroying it, I could completely eradicate its existence from the world, and that it would never come into the world again(at least, not in my lifetime), then I'd probably burn that shit, because I like having a properly functioning world without any sort of godmod powers breaking it.
But if there was a risk of another one appearing in my lifetime, or other people may or may not have one, then I'd probably hold onto it, because the world is pretty much fucked(well, humans are anyways) just by this thing's existence, so I may as well hold onto it for security. I don't believe that I would be someone who could just use it for "the greater good", and I think it would be impossible to not become morally corrupt trying to uphold life/death judgement on the entire world based on my own moral compass. So it would pretty much be used for personal reasons, not for some "greater good TM".
That being said, I can't really think of anyone that I'd actually kill with the note. I probably wouldn't kill terrorist organizations and such because I don't think myself qualified to properly pass judgement onto them as to whether they are "evil" enough to deserve death, I probably wouldn't use it on enemies or anything, because that would be pretty damn petty and I wouldn't consider how they treated me personally to be worthy of death, even if I hate them. So if enemy nation/organization leaders and personal enemies are out, then since I probably won't be killing criminals either, that pretty much just leaves me with people who wish to cause immediate potentially lethal harm to me, or people I care about. That'd still be pretty difficult though as I'd need to know their names, which doesn't seem that likely.
Hm, well I guess I really can't see myself using this thing much, if at all. I'd be a pretty damn boring harbinger of death I guess. I guess it's because I don't really think that anyone is undeniably deserving of only death, because everyone has the capacity to change, and I would be denying them that opportunity by killing them which would be a failing on my part.