Gardeners tend to not want shavings in the manure for gardens. Shavings do rot down eventually, but take ages. Straw heaps do take up much more room, but are more attractive to people wanting manure for their gardens etc. Hempcore, like Rosemary said rots down and goes into a good compost as do one or two others - Bliss is one that springs to mind. When I get a big heap of shavings, I put it under the row of confer trees and leave it to rot down in its own time!!
If you are using rubber matting, you will certainly not need a lot of shavings, so should not have a major problem of a big muck heap. The best thing to do, is just put a little pile of shavings in one corner.