Moss only grows on lawns where there is extremely poor drainage and you can only solve the problem by improving drainage. This can be done by regular spiking and filling the holes with sand, or for extreme cases, introducing some form of drainage.
The 'damage' is almost certainly caused by moles, but by their action they are probably aerating your soil and subsoil and making it likely that the lawn will improve. Just fill in the holes and reseed them in the spring when the damage will reduce once the soil begins to dry out.

Don't be fooled by badgers, Buttercup - we had a lot of problems with them this summer, and the first signs were small, shallow holes scratched in our lawn. They then moved on to the tubs and flower beds and did a LOT of damage, looking for food and moisture in the very hot, dry weather we had. We are certain it WAS badgers, because we saw them. However, I imagine it might be more likely to be foxes in the UK, and they certainly won't be searching for moisture! The general consensus on this site has been that male, human urine sprinkled in the relevant areas is as good a deterrent as anything against marauding mammals of any sort. I don't expect the fact that there is a lot of moss in your lawn has any bearing on the matter, other than to say that I wouldn't expect poorly-drained lawns to have too many worms in them, which is what foxes would be after.

Thanks both of you. We have had moles before in our last house but they always came up through the soil. This one has made no runs at all. It seems likely that whatever it is comes through from the adjacent field. Perhaps it is a fox but what would a fox want with moss?

I think sometimes foxes, like dogs and some cats, when they have marked their "territory" by peeing on it, will scratch the ground quite vigorously: they aren't actually trying to dig anything up.

Check for leatherjackets - if the turf is moist (which is probably is) leave a sheet of polythene over the suspect area at night. Lift it in the morning, and if its leatherjackets, you should see grubs on the top of the soil. Best done when the weather is not freezing overnight though. Starlings and other birds peck up the grass early in the morning trying to get at the leatherjacket grubs.

The other possibility is squirrel damage - they're busy burying things at this time of year, and do make an unbelievable mess while they're doing it.

We have foxes digging for worms....they would dislodge the moss while digging. I have very little moss and they still dig.

I think Bamboo probably has the right answer - I forgot about squirrels: we don't get 'em here.

I've seen this in several lawns and suppose it to be foxes looking for grubs and worms that are living in the dampness. Otherwise birds are doing it before you get up in the morning.