One that's sorta-neat and I'm pretty sure is an accident is where trees spawn their roots below the surrounding ground level, and the generator builds the dirt up around them so the tree has its own little mini-cave and just has branches sticking up above ground. Especially when the dug-out area has mushrooms or glow worms in it, that makes it look intentional.
Other areas could be things like little caves of mini moon-dwellers (shrink the normal sprites down to %10 their size and have them only stay in the cave area, maybe remove their dialog and have them weakly attack any critter that comes near except rabbits). You could also find the occasional cave or hole where a single pathetic human is hiding (do you have an animation for humans sitting with their arms wrapped around their knees?)
I like the hound hole idea, and the different well types. What about the spring I mentioned as a bug? There could be unique plants growing out of it that draw specific minerals from the fresh spring they can't find anywhere else. What about a poisoned tree grove? The trees could give off those status effect particles to show they're toxic. Some sort of animal could be immune like birds. Also maybe a certain armor set makes you immune too.
Some sort of mushroom that grows in the world should trigger that weird shader effect for a minute or so