Kinda like green onions, the white bulb part is the strongest and most desirable, the further up you cut gets into the green part. Depends on what they look like. If they're fresh and robust, I'd cut them all the way up and just discard the furthest green tips. They're small so if you're frugal, you'll probably want to use as much of it as possible. I don't think the greenest part is stringy or fibrous in any way.
I'm guessing you might be able to freeze them but I'm not sure. When I got them from the Vets club, they were soaking in a big tub of water.