Your Shag Moss looks like a Brachythecium species. This is a fairly robust genus with species enduring some direct sun. turning white is one character that appears associated with it.

Your best bet is to grow it successfully and replant portions in your yard.

Not all that many moss species are available over eBay.

Thanks for the response.

I've been looking at these and there's just too many species :)

You're probably right. My main issue is that I'm having trouble growing them from small pieces due to all the weeds, leaves and other debris. Maybe I'll trying growing it on a covered substrate and transplanting

I am awful with moss ID - I adore it, but can't tell one from another. My tip would be to keep your eyes peeled **everywhere**. I keep a box, bottle of water, paper towels, and a wide-blade putty knife in my car all the time. If I see a sunny parking lot with moss... well I go collecting.

I have never had anyone complain about me taking parking lot or sidewalk moss. (obviously don't help yourself if it looks like someone is purposefully tending the area).

There is one very, very green, low growing, clumpy moss I have seen growing in a lot of parking lot/sidewalk cracks that seems to take a lot of sun.