Having lived most of my life in cold conditions, I've never liked nor used a "trickle" of water running. It may keep you're incoming water from freezing, but what's going to happen to the outgoing water?
The incoming water is not going to be much above freezing. Then this trickle runs down a cold drain pipe and that water that may have been a few degrees above freezing, is now going below 32* and freezing in your drain pipes closed..........
When you run a volume of water - wash dishes, flush the toilet, take a shower - something more than a trickle, the outgoing water warms up the drain pipes due to the volume of water, lets all of the water drain and there's no problem.
Bill
Nodwell RN110 out moose hunting. 4-53 Detroit, Clark 5 spd, 40" wide tracks, 10:00x20 tires, 16,000# capacity, 22,000# weight. You know the mud is getting deep when it's coming in the doors.