I'm lucky. I live on a dairy farm so we spread the horse muck with the drier cow muck. As it's not stored for any great length of time, it's the main reason (aside from cost) that I use straw and not shavings.
Some of ours gets taken away to be made into bricks. We actually get some bricks for it. The rest (shavings) gets spread on the school to make it more cushiony. Works a treat.
Our muck heap is a trailer (we tip onto it) and it gets taken away by my YO/ YO's OH and dumped on local farmer's fields. Where it stays for a couple of years until he finds time to spread it (farming isn't his main job!)
We have a seperate field heap, pure poo (and a bit of hay from winter), which gets taken by the local gardener person thingy (can remember his name, just not his official job...)
Our muck heap is actually a purpose built trailer attatched to the back of a tractor. When it gets full, we drive the tractor down the lane (which can be interesting as the brakes are very tempermental!) and dump it down the bottom, behind the old railway. The local farmer sometimes takes some for spreading, but otherwise we let it rot down and if we get a very long hot summer, we burn it.
One of our neighbours has an allotment so he takes it away once it's rotted down and it gets used by him and other 'allotmenters.' Some also gets put on our garden
I keep my heaped up high and ensure that in winter it is always simmering away and burning down. In the summer unless we have a windy couple of days, I just leave it, but then all the winter muck (the bulk) has been got rid off.
We had our muck heap ignite on its own one summer and because the paddock was terribly sun-scorched the remaining grass lit up as well. That was terrible, so now in the summer I check it and if necessary give it a hose.
All the muck at our stables (26 horses) goes up a ramp onto the back of a huge trailer & is taken away daily & dumped in a field. A farmer uses it later in the year on his land.
ours is currently loaded onto a trailer as cant get in the field to empty wheelbarrows, its then taken to the farmer where YO buys her hay and dumped on his fields for him to spread