Creator Breakdown: Rumors abound that several of Andersen's works are a result of dealing with his own romantic and sexual issues. The Nightingale was allegedly a tribute to a singer known as "The Swedish Nightingale", Jenny Lind, who did not reciprocate Andersen's romantic feelings. "The Little Mermaid" was a similar case, of Andersen dealing with 'losing' a close friend (one he had feelings for) to marriage.
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."
A number of Andersen's stories end with the protagonists dead, but this is considered a good thing because they are going to a better afterlife, leaving the miseries of this world behind.
This is most notable in Japan, where adaptations of his stories crop up every few years. The Little Mermaid fit very well with Japan's own history of Mermaid stories (though Japanese mermaids are much scarier). The Little Match Girl is often referenced in other works.
Americans like Andersen's fairy tales too… but often are only familiar with the Disneyfied versions.
Harsher in Hindsight: Pretty much his entire canon really, particularly "The Little Match Girl". Also "The Red Shoes" – in this age of consumerism and 24-hour advertising, a girl condemned to a painful death for wanting a nice pair of shoes can be outright Nightmare Fuel.
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. It has one evening of Christmas glory... then it is stuck in the attic to slowly die, 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...