The topic that Mauro started gave me an idea...How does the uneven terrain and surface changes your stance while slinging. What are the "sacrifices" that you have to make?
Let's start with snow covered flat surface (that is the case with Mauro and Timann and other snow slingers)... Like I said in snowy surface i tend to use wider stance with bended knees (fig 8) and smoother torso twisting...

I simply use arm movement only and a little hip twisting on bad terrain, nothing elaborate.

Last night, I adopted the same position Dilyan described here and in the other topic.
I still have to try slinging at a target placed higher or lower than me, but as an archer I did that many times.
In that case, the position was pretty much the same, but I don't know how much could this help with slinging!
Greetings,
Mauro.

Same as kid just there's no run up or anything like that. just get comfortable and make sure you keep your hip rotaion and ususaly there isn't a huge difference in power but there is a definite noticable decrease in accuracy. But like slinging on notmal ground it's probably a practice thing.

One of the places I like to sling while walking the Huskies is a local gravel pit that was converted  into a park.  There's a trail that goes around the top of the pit.  That's where we usually walk.  When I want to shoot at targets on the bottom but close to me and consequently more straight down I find this helpful; move the rotor behind your left shoulder (if you're right handed) when winding up.  This will take almost all of the arc out of your shot.  You'll have a sharp, hard, line drive type shot, making it easier to hit objects lower than you.  Try it, you'll like it.

I've thrown in some pretty rugged terrain, and until now I've not really put too much thought into how I maintain my footing while I sling. If I'm on the side of a steep slope, I have a tendency to crouch a little bit to lower my center of gravity. I won't step into my throw like usual and generally keep my feet planted. I try to keep everything else the same.

I think a bit of the opposite is true. Consistency brings accuracy so the more your throw stays the same the better off you are. It works the same with marksmanship of any kind. Archery, guns, etc.

And what about the thickness of the sling. Mine sling are usually no thicker than 8 mm in diameter stiff braided

It doesn't really matter. Slings can get too thick at a point but that's pretty thick. Look at the Balearic slings. My Balearic slings are nearly 3/4 " wide and 1/4 " thick (flat braided) coming out of the pouch. Thick braids need a tapered release cord to release cleanly. Just experiment with all these variables, and when you find what works, go with it consistently. Everyone does these things slightly differently. As long as you do it consistently and it works for you, it doesn't matter.

My problem with throwing on rugged terrain is maintaining my footing...which is why I tend to crouch down a little bit to lower my center of gravity. The style or thickness of the sling is a non-issue...I just don't want to end up falling down the slope.

I saw this post soon after registering and wanted to reply.  Now that I have thought about it more, here are a few (unoriginal) thoughts:

You first asked about snow.  My preferred throwing style is underhand or Greek.  In shallow snow (say, up to ankle deep), no other adjustments are necessary.  In medium snow (say, ankle to knee deep), underhand is now compromised because my slings would hit the snow at 120 degrees from vertical (around 4:30 on a clock).  Greek becomes necessary.  In deep snow (knee to waist...or--gulp--higher!), slinging would be...would...what in Hades am I doing out here, I can't even walk!

Underhand also has its compromises in dry (non snow) terrain.  Here I am speaking of terrain features like tall grass, young tree/bush shoots, rocks, etc.  Even some thick grass can instantly decellerate my sling's downward rotation and spill the pouch, or at least throw off my aim dramatically.  If I don't have a relatively smooth space to sling, underhand gets difficult, and I tend to gravitate to a low sidearm, high release figure-8, or my most accurate option, Greek.

My choice of underhand is due mostly to a rotator cuff injury from lifting weights in my 20's, mixed with advancing age which makes overhand or sidearm painful after about 20-30 good shots.  It is much easier, safer, and just as effective for me to throw underhand under the right conditions.  However, we know that ideal conditions aren't always present when you feel the need to hurl a projectile.

I like Greek as an alternative since it combines the baseball pitcher's maximum windup with a predictable release point.  That said, I am captivated by the "controlled release staff sling" concept and plan to make one shortly.    Having fished for years, I am convinced that with the right setup, and the right balance of materials, I can get speed, power, and accuracy--if perhaps slower reloading speed and less portability.  If I can place a one-ounce spoon in a one-foot target at sixty feet with a fishing rod (fast or slow), I am confident with practice that a 2-3 ounce rock would do the same...and now I won't be afraid of scaring any fish!

Sorry, I digress.  Terrain features that would slow down the staff sling would also include bushes, tree branches, rock ledges, or anything else that would spill the pouch.

Would that be a slinger's epithet?  "Spill your pouch, dude!"

Some members here don't care for underhand.  There are certain situations where, due to obstructions like branches,  underhand just makes more sense.  Also, for me it's by far the most accurate style.  I guess because there's only one plane to worry about.  If you line up correctly with your wind up you only have to worry about up and down.  

On gravel, dirt, snow and sandy terrain I too have to use a lower and wider stance.  But I always spin a little on the ball of my foot when I am at the final release of the throw.  It ends up creating a divot, and then I have to move to a clear spot to sling again.  If I don't change up my footing to a clearer spot, I notice that my accuracy is affected.