all the time. Rocks with a flater profile will buzz as they exit the sling. It happens with just about any style cast. The backspin on a rock is a fun thing to watch. If the rock is just right and the backspin is oriented vertically the rock will look like it's hovering. I've actually had them drop to about 3 feet off the ground then gain altitude and float at head level for 100 yards.

Yes, I have heard it, specially when I throw the rock with an underhand shot.
 
Chucker
If you were talking in plural, I like the rounded ammo the best because it is 100% aerodynamical and it never tends to change the direction where you shot it.
 
Jandro A.K.A. Ra

Welcome all new members!
 
Usually I don't like the "singing shots", feel them as muffs due to both poor distances and accuracy. But sometimes they give a beautiful effect, indeed. Most often reason of it is in an unclean release or poor shaped stones or both together. In these cases stones can fly along very weird trajectory and each of them gives own peculiar melody.
 
Sometimes granite stones brust during hard shot, especially elongated ones. A few times I had the case that stone exploded to two or three pieces after flying some distance.  
 
Jurek

Remember, if you use just rocks you find that are irregular, they WILL tear your pouch very quickly. Hear in Arkansas I usally throw the irregular rocks. So, don't take any sling that your are to proud of, like a braided sling.

Well, I can get into this post; boy howdy!!!!
 
Yes, indeed, the rocks sing for the very same reason bullroarers roar.  In fact, the sharper the edges on the rock, the louder it will sing.
 
Musically speaking, a bullroarer is known as a free aerophone.  This means that it causes the air to vibrate outside the instrument.  The bull roarer does this by spinning fast enough that the air vibrates as it crosses the edge of the blade.  When a stone spins fast enough and has a good edge or two, it, too, becomes a free aerophone.
 
When I was just a boy, I would shoot rocks from my Saunders Folding Falcon slingshot.  My grandmother lived on a gravel road so ammunition was plentiful, but rarely round and smooth, even though it was river stone.  Over the years of being driven on and having some chert mixed in, the stones tended to have sharp edges.
 
I would sometimes shoot stones onto the road at about 30 deg. down from horizontal.  When the richocheted, the sound was as cool as it gets.  Indescribable in text except that it was obviously caused by the spinning stone.  Certainly it was doing several thousand revolutions per minute.  A really good one would sing for a hundred yards until it went into the woods.
 
Jeff <><

Cool!
 
I used to make bullroarers for craftfairs.  Sold like hotcakes as unusual Christmas gifts.
 
Based on your description of the theory of operation the optimal "singing stone" would be flat, thin and have sharp edges.  I think that pretty well fits with my own experience.  Unfortunately this is also the shape of stone that is most squirelly when you throw it.
 
Have you ever experimented with what variable controls the pitch?  I suspect it would be the width but always made my bullroarers one-size-fits-all.  Seems like rotation speed might play a roll as well.
 

The real fun begins when your aim reaches the point that you can send those 'singers' about 10 feet overhead of your target.  I've done this several times while camping.  After you've send a few of them nicely humming over the picnic table area in the next campground over... then try to convince the happy campers that you're a scientist from a local university, studying the effects of prozac and caffeine on humming birds -- ask them if they've seen or heard any humming birds that seem to be behaving abnormally energetic etc....  
 
Just be sure your aim is good enough by this point that you won't send any insanely-happy humming birds through a tent or car window!