My mother was one who ALWAYS dropped her threads on the floor.....and I got the pleasure of sweeping and cleaning them up (NOT!). SO I have always made it my business to cut threads and put them in something.
I have a paper lunch bag that I folded the top down on and use blue masking tape to adhere to my table, while in class. At home I use a small plastic trash can with a liner that sits beside where I sew.