Finally, I looked on Pinterest for a solution. I found one! I made a little fabric can to dangle over the side of my worktable, held in place by a wide ribbon glued around a baked-potato-sized rock. Nifty, huh?
The thread catchers I saw online had pin cushions to keep them in place, but I subbed in a rock because I have a pin cushion I'm devoted to (Genevieve reminded me that she picked up the rock on this blissful vacation last spring).
I did try to sew a sleeve for the rock, but then I thought the trash can would be more portable if the rock was firmly attached to the ribbon. I wrapped the ribbon around the rock a time or two, hot-gluing as I went. I intend to carry the little can to the living room coffee table or dining room table when I carry my handsewing there. No more sewing snips on the floor!