You will always get moss unless you aerate the ground thoroughly and give a good 1/2" of horticulture sand all over and sweep it into grass as soon as this very cold weather is over.
It's fine to put it in the compost bin - I've just scarified our lawn of moss, but for the moment I'm leaving a pile of it on a bare patch of earth as it's about to be nesting time and the birds will probably be very grateful to have some of it for their nest building.