Do you bother to do anything with the moss or just accept that at least it is green?
Usual cause of moss is poor drainage coupled with an aspect which doesn't help the lawn dry out (i.e. yours by the sound of it.) Having said that, my lawn is south facing, well drained, but very mossy - my theory is that the soil is so low in nutrients that the moss finds it easy to out-compete with the grass.
Manual moss removal (very hard work without a powered scarifier), aerating the soil if necessary (also hard work), regular feeding if the soil is poor (expensive).
Guess what? - I've learned to love my nice, mossy, springy 'lawn'!