We have started with a carport that is open on all sides. As we have noted the direction we walk to it from the house or garage (where hay is currently stored) and the direction of the sun (more concerned with keeping things cooler since we are in Florida) we have decided to add some walls (with gaps between boards to allow air flow) and have the opening be to the east so it will provide afternoon shade. Luckily, at least for now, the neighbors big water oak in their cow field provides some nice morning shade 
Mine are sheltered by the barn itself and the side of a big hill, so they are open on all four sides. If I was going to close up the back/sides, they would face SE to get relief from winter wind and late afternoon summer sun.
Upper Midwest. My shelter opens to the SE.
Upper Midwest. My shelter opens to the SE.
I am in Ontario, and all of ours face south. Most of our winter weather seems to come from the NE. So that works for that.
However, being open to the south means that the sun shines directly in on the hot days.
I do have trees, and the horses also stand in the shade of their sheds. But it is not perfect. I keep thinking how I can fix this.
The typical answer is ‘south’, but this is a situation where you should use your brain and apply it to your location. It also matters how the shed is used and whether they can access all the sides.
In my particular location, nice breezes come from the west and storm winds usually come from the west or the south. In my particular location, I am sheltering against rain, wind-driven rain, and sun.
South means that the sun can get into your shed. This is the right choice if you are most concerned with keeping it dry and warm. This may not be the right choice if you are concerned about hot weather, especially if the shed isn’t very deep or if horses cannot access the back and use the back side as a sun block.
The sun’s angle changes during the year, to be nearly overhead in summer and to be low in winter. This means that in winter and fall, the sun will get further under the overhang than in summer.
The most brutal sun in my location is from the west on hot afternoons.
I tend to face my structures east and north.
We have one east (on north side of barn - not ideal but it works in all but worst of east winds as its deep) and one south.
Plans for south east for next run in.
In our area, a North facing shed would spend all winter full of snow piled tight in there, that would not melt until next late spring.
We have North walls in barns that we have drifts on them all winter.
The OP needs to decide what weather she is providing shelter for in her area is and go from that.
Maybe look at how others there face theirs and ask why?
In addition to the winds, consider runoff. And to a much lesser extent, I consider horse visibility.
I like to, when possible, position them so the water isn’t going to run into them, away from prevailing winds, and so that I can see the horses eating in them when I stand in the barn. So mine are currently open to the East, but one has a bit of a East-NE tilt that doesn’t effect its efficacy, necessary due to the ground I had to work with. Neither gets more than a dusting of snow inside even when we have several feet. I watched the snow patterns around my barn for a year or two before building so I was pretty comfortable with what it would do drifting-wise on this property.
Here’s what I learned while boarding at different barns:
Run in sheds that are placed far away from the barn are not used often as horses want to be near the gate/barn/house.
Run in sheds placed by wooded areas where the wind causes noise in the trees are not used as much as run in sheds placed in areas away from trees.
Run in shed should face away from the prevailing wind and sun so that in summer the horses have shade. And so that the wind does not drive rain into the shed head-on.
I’ve seen great looking run in sheds that horses did not use because of the location away from the area closest to the barn.
I’m facing having to put in shelters instead of a barn* and have decided to put window openings on three sides and a door opening on the other side. I intend to put drop-down coverings on the window openings and possibly a sliding door on the door openings. The window openings will allow the horses to see each other & let me feed the horses without going inside the shelters. The prevailing winds here come from the southeast (for rain) and northwest for other weather. Except when it doesn’t.
*Our county requires engineering papers for anything over 120 sq. ft., but structural engineers in the area have backlogs like I can’t believe.
My big shed is 30 ft long, 10 ft (12 ft roof) wide, situated so that the short ends face west and east. The west 1/3 has walls on the west and north sides. The east 1/3 has walls on the east and south sides. That way, no matter what direction the sun/rain/wind is coming from, there is shelter.
In Midwest mine face South and Southeast.
