Because I dont have a great deal of silk floss I store mine in a special box. The leftover thread I store in small bags, with the thread tag, so I dont get muddled up.. Be interesting to see how other stitchers store their silk floss....
If it came as a skein, like six strand cotton, Id use a stitch bow or a baggie. Id be reluctant to wind it on a bobbin because its difficult to get the kinks out later.