I'm surprised no trolley users have chimed in.
For me, the putter cover either goes in the slot where the putter was, or in my back left pocket during a putt. Sometimes on the ground if it isn't sitting in my pocket in a nice way.
For driver/wood/hybrid, it usually goes in the mesh carrying area on my trolley. I often can tell that I am going to be taking a 3W or hybrid for my upcoming shot, so I pull the club out of the bag, remove the cover, and place it in the mesh area all before I get to the ball. Make stroke, then start walking while reversing the process in motion. Sometimes if I can't tell that I am going to use one of those clubs, the first part is done over the ball, but I always put it back on while in motion.