I hang the saw almost all the time, from the tiny ms 150T to the monster 3120xp.
I stow only to avoid dragging the chain on the concrete, gravel, roof, wires or groundy's head. But I don't like it because that adds a parasitic weight / movement near my butt and that disrupts my own movements/feelings. Moreover, I hate to have the sharp teeth in reach of my legs and/or in the way of my right arm and hand (bloodied glove as an evidence in my first climbing week). The last point: if the rope is caught around the bar (descent or fast rewinding the tail), there's almost no time to react to prevent an hard snug and rope's damage.
I dislike the commercial bungy lanyard though for two reasons : it's too short for me to allow a full reach (over extended arm toward the left with a side way cut) and I have to sew an extension, annoying. Two, the recoil effect from the internal bungy adds to the saw's weigh and hampers the precise positioning.
So now I use one of my retired multipurpose 3 strands slings. Strong enough for the task, even if damaged, cost nothing, gives something to grab between the fingers, I don't cry if I nick it during a cut and I can pick a "new" one at will. No problem of supply.