My knowledge on burls is somewhat limited, but I'll toss out a few answers as I understand things, and let others correct me if I'm wrong.
Despite the dictionary definitions, as I see it there are two kinds of burls. One is the true burl, which is a growth on the trunk or a branch of a tree. I've heard it referred to as "tree cancer" since it's an unruly group of cells growing abnormally, resulting in wood with a wild, varied grain pattern. The other is a root burl, which is generally the twisty. knotty, wild-grained portion of the tree at or a bit below the ground level (depending on the particular tree). So to my way of thinking, "burl" wood can be from either of these two sources. There is often other wild-grained wood in the crotches of trees, too, but I wouldn't consider it a "burl".
Assuming you're asking about a true burl, their rarity will vary from one species to another and one location to another. In some forests they can be nearly impossible to find, and in others they'll be more common. I have no idea why it varies, but just that it does. The very large ones are indeed rare, but I'll bet if you walked around most forests looking for burls, you'd see some small ones pretty quickly.
A lot of burl wood is used in flatwork, but it's often veneer instead of solid wood. Burls are usually too small mill lumber of any substantial length from them. But a single board foot of lumber can make quite a few dollars worth of veneer. I've also seen pieces of solid burl used on smaller work like jewelry boxes and such. Some burl is very stable as it dries, and other species move a lot. The wild grain can make the movement pretty unpredictable. Burls also vary quite a bit in hardness. Cutting maple burl is easy as butter compared to something like mallee or manzanita burls. Some burls can also be pretty punky and rotten inside, especially if they have been on a dead tree for a while.
I believe removing a large burl from the trunk a tree will almost always kill it. I would think one growing on a branch could be removed (along with the branch) without killing the tree.
That my