It really depends on the kind of snow. If the snow is dry and fluffy, I think you need at least a foot. If the snow is wet and clumpy, at least eight inches. Sometimes you can get away with less if there's a solid frozen top on the snow. It all depends. If you're shoeing on a small snow fall, I'd try and do this on a dirt trail - not on concrete. Hope that helps.
For me snowshoeing is a way to travel while I go out and do what I enjoy in the outdoors. If it is easier to walk in shoes/boots, I do, so it probably takes a foot or better before I deem it easier to get about on snowshoes. And again, that varies with terrain and type of snow.