These are what we call the 'YMMV items.' Things that some people find in this work. We call them 'your mileage might vary' because not everyone sees these things in the same way. This starts discussions in the trope lists, a thing we don't want. Please use the discussion page if you'd like to discuss any of these items.
Esoteric Happy Ending: In The Cripple, a paralyzed boy manages to drag himself out of bed and attempts to walk so that he can save a caged songbird from a cat creeping up on it. Everyone celebrates the child's improved state, but in true Andersen fashion "the bird, of course, had died of fright."
For a modern viewer, depending on taste, it can be very easy to read The Emperor's New Clothes as a condemnation of certain types of modern minimalist art.
Nightmare Fuel: There's a story about a fir tree that dreams of being a Christmas Tree because of the honour and the glory. Getting cut down hurts like getting your legs cut off. Then it slowly dies, like Christmas Trees do.
The hellish punishment in The Girl Who Trod on the Loaf. The Red Shoes, where the heroine is only saved from damnation by having her feet chopped off. And the literal nightmare in Aunt Toothache, featuring the Anthropomorphic Personification of aforementioned ailment, who has drills and tongs and similar implements for fingers. It is possible to go on...
Tear Jerker: Andersen was a master at tugging the old heartstrings.