A trick I learned from my seamstress wife. Carry a lighter with you, and test burn a thread if you are not sure, linen/silk/wool (natural fibers) will be hard-ish to get started, and will mostly stop itself. The man made fabrics will burn easier, quicker, and curl into hard little ends. Silk does the curling a bit too, but the burnt end can be separated with your fingernails, whereas the plastics will not separate so well. The smell is different too, the poly/rayons have a very plasticky, acrid smell, and the natural fibers smell like burning hair a bit.
If you are buying new fabric, it should all be labelled, this is what we do at the thrift stores, where you never know what is what. Hope this helps.
Mike