UPS selection: Least electrical overhead?

As far as i know they only use power when they are charging the batteries. I may well be wrong tho.
Varies by model. They should be trickle charging the batteries in normal use. Better ones will actually supply power from the batteries and thus always be charging. In any case, we aren't talking about enough additional power usage to worry about. Most people have numerous random devices around the household "wasting" power.