This plant is not exactly a moss, it is a liverwort; they are related but the differences are quite technical and beyond me to even attempt to understand. If I had to make an observation I would say liverworts have waxy, leafy structures whereas mosses seem more flexible and slender. I will say no more as I expect there are specimens of both mosses and liverworts that blow that theory out of the water!
This particular species is known as the crescent-cup liverwort (Lunularia cruciata). It is best known as a coloniser of flower pots, rockeries, walls and garden paths. In some cases it can be of a pest in gardens. It is this fondness for gardens that lead to a belief that this is not actually a native species but was brought in with imported plants and has spread as people have bought pot plants from garden centres. That may be true but it can also be commonly found on stream banks and bridges and it can even occur along woodland rides. Whether this is because the plant has escaped from gardens or because it occurs naturally is not clear.
The 'leaves' are about 1.5mm across, are pale green and almost always form large spreading mats.Lunularia cruciata: the crescent-cup liverwort