And then of course there was the horse. I don't know how hard it is - or if it even is possible- to use a slingshot on horseback, but I never heard of any examples while horse-mounted archers, from the Mongols to the Plains Indians, are well documented.
Slings were on occasion used on horse-back but only on "special occasions" and even less in a warfare setting, they were used as back-up weapon.
One example would be the people of Tibet, they didn't have much in the way of wood so slings would be used for hunting and herding.
The reason that you wouldn't see it used often is not only because Slingers were generally considered to be the lowest rank of the army thus they didn't have a horse normally. But it was also because slings need a good center of gravity and some predictability of movement.