Stumps can randomly generate many little goodies. Forgotten cobwebs, chestnuts, smooth stones, dried rattler skin, singing old logs. But likely if you're reading this you gave up long ago when you first started looking for these fabled treasures. The question is why can others find these but you didn't?
As far as I can tell, anytime a map is loaded empty stumps get a chance to spawn goodies. It is fairly low however (as in 1/50 at the very highest I'd guess), and when everyone was first wandering the world they had the 1/50 chance to get toys. Most didn't and gave up. Stumps surrounding active villages (2-3 maps out) will however get a lot of map loads as foragers head out on their journeys. This means those stumps may have as high as a 1/5 chance of having spawned something by the time you get around to checking them (if others aren't). When wandering my village's foraging grounds, I found 7 snake skins, 5 cobwebs and 1 singing old log after checking about 50 stumps. If anything it makes snake skin too easy to get at current spawn rates.
Short story? Check stumps near high traffic areas regularly as they will have more goodies. And don't destroy stumps 2-3 minimaps out from your village as they will be a major and helpful source of curiosities and snake skin. Alternatively, randomly foraging and finding lots of stumprises likely means the area is populated nearby.
I'm not sure I like how these spawning mechanics work, but at least it's a benefit to heavily foraged areas which otherwise tend to suffer in quality.
Edit: This is of course simply my observations. I could be wrong on how it works.