Fungi turning up is a good sign. They arrive where the ground is undisturbed and unfed, which is exactly the condition moss wants and a lawn cannot tolerate. No mowing, no feeding, no chemicals, and this is what you get instead.
One of the pleasures my wife and I share, is strolling through the moss garden after rain showers have awakened the fungi. Spotting the colorful but brief visitors and admiring their presentation in the ocean of green mosses.
When detected, I often ease their eruption by parting the dense moss growth which can be strong enough to disfigure the determined sprouts.
Sometimes their debut is over by the next day, their glory shortened by a hungry squirrel, but often their persistence is an allure lasting days as their hues shift with maturity.
Mushrooms are the fruiting bodies of mycelia, fibrous primitive soil dwelling hyphae. They are great contributors to the soils, increasing the efficiency of water and nutrient absorption for most plants and aiding in decomposition.
Gardeners are just beginning to appreciate the role of mycelia in their soils, and some now buy powdered mycorrhizal fungi as a soil additive. Moss is the exception that proves the point: it forms no mycorrhizal association at all, takes its water and nutrients through its leaves, and needs nothing added to the ground it sits on. Perhaps the long connection between toadstools and the folklore of Faeries and the Moss People are the most intriguing. Sparking the imagination of young and old is as important to encouraging good land stewardship as anything.
One had a lovely face,and two or three had charm,But charm and face were in vainBecause the mountain grassCannot but keep the formWhere the mountain hare has lain.~ W. B. Yeats
Undisturbed, unfed ground is what moss wants. It is also the cheapest ground to keep.

