From more flammable to less flammable

cotton/linen
Burns with a hot, vigorous flame, light colored smoke, and leaves red glowing ember after flaming stops. Does not melt or draw away from the flames.

rayon/lyocell
Burns similarly to cotton and linen, except that it may shrink up and become tighter to the body.

acetate
Burns with a rapid flame and melts when burning. May melt and pull away from small flames without igniting. Melted area may drip off the clothing carrying flames with it. When flames have died out, the residue is a hot, molten plastic and is difficult to remove from any surface.

acrylic
Burns similarly to acetate, except that it burns with a very heavy, dense, black smoke. It drips excessively.

nylon, lastol, olefin, polyester, and spandex
Burns slowly and melts when burning. May melt and pull away from small flames without igniting. Melted area may drip off clothing carrying flames with it but not to the extent of acetate and acrylic. Residue is molten and hot and difficult to remove. May self-extinguish.

wool and silk
Burns slowly and is difficult to ignite (especially in winter garments). May self-extinguish.

modacrylic and saran
Burns very slowly with melting. May melt and pull away from small flames without igniting. Self-extinguishes.

aramid, novoloid, and vinyon
Chars, does not burn.

Sources:
The U.S. Department of Agriculture Cooperative Extension Service,
Iowa State University of Science and Technology