I have a trash can right beside me on the floor and I try to put threads etc. in it, but sometimes they land on the floor. At the end of the sewing period, I take a lint roller and "roll" up the threads that missed the trash can.
I feel that I would waste too much time to try to get every thread into the trash can as needed....so I let it go. This didn't used to be...used to be anal about getting every thread into trash at the very moment it was cut.