Moss can be quite attractive as a 'lawn' Dave, but here we often have drought conditions, and the moss would need copious watering in summer or it would die off and go brown. (Water meter - expensive!)
So we remove it, using an electric scarifier. It rips out the moss which we then chuck in the garden waste bin.
Last autumn I bought a box of lawnfeed/moss killer powder and we sprinkled it over both our small lawns. Very good results. Moss all gone and grass bright green. It seems to have filled up the spaces left by the dead moss, so all is well.
Husband can't see the point of a blinking lawn. He'd never seen one until he arrived in UK, and would probably prefer our garden to be a gigantic vegetable plot with a few goats grazing on an enclosed section of it!