10x10 or 12x12 seem to be most common sizes. Higher the better, both for safety and air circulation. I don't think it's rare for horses to lay down. Most ones at our barn lay down every night and my 17H TB has no problem getting up and down in a 10x10.
I think the biggest factor is how much space you've got to work with and how many stalls you want. I.e. if your barn is 60' long and you want the lean-to extend the whole length, that means 10x10 if you want 6 stalls or 12x12 if you can get by with 5 stall.
You don't say where the doors will be (barn side or eave side) or what roof slope you want (mostly picked for aesthetic reasons to complement or match the barn roof). But if the stall doors are on the barn side, and you use a 4/12 pitch, a 12 ft stall with an eave height of 8 ft would give you 12 ft height at the barn wall which is plenty. Can a horse bump his head in that scenario? Sure, but will probably only do it once or twice. :wink:
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