There are more then 15,000 different 'varieties' of 'moss' and moss in general. But I will assume you mean one type and will address it, and I think it is "Leucobryum".
(Just because it produces 'spores' does not automatically mean it is a fungus. FERNS are plants too, and produce spores.)
Look at this photo and maybe this is the type of moss you are talking about:
http://www.biopix.dk/Photo.asp?Language=la&PhotoId=13631
It is not a fungus because it performs photosynthesis.
Moss doesn’t grow flowers, but rather grows tiny spikes that produce spores. Plus, moss doesn’t have real roots. Instead, it has rhyzoids that help them attach to rocks, trees or soil. Because of this, moss pulls moisture directly from above in the air and rain water, which means leaves and debris must be kept off the mats of moss for best growth.
Being a non-vascular plant, moss is more similar to green algae than to plants such as trees and shrubs. Without a method of transporting water up from the soil, the moss needs a damp climate in order to photosynthesize and grow. When the moss dries out, it does not die. It simply goes dormant until water returns. Without adequate moisture the moss will shrink back, lose its green, and cease growing. Mosses also need water in order to reproduce.
First one cleans up the lawn of all the winter's debris, the leaves, old cans and wrappers, and any dog waste that was deposited on the snow. A rapid raking to pick up all the trash and then the moss control is applied.
Almost all commercial moss control is iron sulfate, no matter what the brand name. It turns the moss black within a couple days of contact and makes it ready for removal. For years we have applied commercial grade iron sulfate, let the rain wash it into contact with the moss and thatched it away. In the last few years we have experimented with liquid forms and been thrilled with its rapid action and ease of application.
No matter what choice you make for the substance, a few ground rules apply. It's pretty safe to handle and apply, but is not good for aquatic environments; so be sure none washes into the streams, rivers or ocean. Keep it out of your eyes, wash your hands after applying it and do not get any on your cement as it will stain it black.
The underlying question still remains. What can we do to prevent the moss from coming in the first place?
It turns out that the mosses in the lawn are really a symptom, not a cause, they are an indication that the conditions are favoring them over the grasses, and to change the conditions requires a greater effort. Grasses want good drainage, an only slightly acid soil, steady low levels of nutrients and constant but light mowing.
Then some people WANT to cultivate to make their yard entirely moss.