I have no Spanish moss in my neighborhood, but only a mile down the road the neighborhoods are covered in the stuff. And that's because Spanish moss requires near constant high humidity and those neighborhoods have tons of trees that create a sort of microclimate that traps the humidity -- you probably have a lot of trees around.
What's funny is that I mostly only see the Spanish moss growing on Live oaks and they stay away from most other trees, so I don't know why you have it growing on mulberries and other trees in such heavy quantities.
I would just rip it down and use it as mulch/compost. BTW, it's not actually a moss, it's actually classified as an Air plant, because it has no roots and it gets its nutrients/water from the air.