Step 1: Garden hose bobbin holder
My solution was to take a piece of garden hose that's 5/8 inch (16 millimeters) in diameter and split it lengthwise all the way down. The bobbins then snap right into the hose and stay very secure until I'm ready to use them, even if I drop it on the floor. The sides of the bobbins are always facing out, so I can easily tell what color thread is on them. An added bonus is that the hose has a slight curve to it, which keeps it from rolling off the table.
I was lucky enough to find discarded garden hose, so it didn't cost me anything. But it would probably look even cooler if you got clear tubing and used that instead.