First post :0)
Many of my customers had 80-90% moss lawns despite having regular gardeners before me.
I think the previous guys used to just tickle over the top as the sponge layer in some gardens could easily be 4-5 inches deep!
My view with a heavily mossed lawn is to cut it moderately low on the first cut and remove masses of moss and even lower on the second cut a fortnight later untill you can feel the ground beneath your feet rather than a moss carpet.
You must have a powerful mower though or your machine will sink and keep cutting out. This can really stress an underpowered mower.
Otherwise scarify the lawn which is offered as an extra service. If you only service domestic lawns you can pick up an electric scarifier really cheap.
Hope this helps.