I'll admit that the biggest draw to this story for child me was the abundance of magic. There are genies and magicians and tricksy cleverness aplenty. But it's also a valuable story in that it demonstrates the necessity for wisdom in judging the character of others and in what you wish for. Not only that, the story places as much value on hard work and (depending on your version) honesty as it does having a magic class-crossing genie in your pocket.
Hansel and Gretel
I'm not really sure why I wanted to include this one; it's one of the darker fairy stories, even in the sanitized versions. Maybe it's the image of a house made out of gingerbread. Maybe it's Gretel's intelligence saving her brother. Maybe it's because it gave us the Looney Tunes character Witch Hazel, the exquisite running gag of "Hahn-sel" (like Hans, as opposed to "Hansel" like "hands"), and the wonderful insult, "Ah, your mother rides a vacuum cleaner." Who knows? But it is a fun story to come back to with different elements emphasized or explored.
Little Red Riding Hood
The Snow Queen
My penchant for "The Snow Queen" stems from two Hallmark-produced miniseries, one based on Andersen's fairy tale and another based on "Snow White" that borrows elements from Andersen, namely the shattered mirror and its effects on those who find a shard of it in their eyes. It's a beautiful tale of love and friendship conquering everything from emotional coldness to laziness to outright wickedness. It also features one of the best heroines in all of fairy tale lore: Gerta.
"Bluebeard" is a weird story. It features locked doors, prohibitions, and mysterious bridegrooms (a bit like Cupid and Psyche, but not so happily ended). While its original is most likely a cautionary tale about curiosity (a la Pandora) its variants feature the bride saving herself as well as being saved by brothers, sisters, and mothers. It's a creepy story but worth the read if you find the right version.
Here's another tale rife with magic, but the magic isn't necessarily what brings me back to this story again and again. The things I love about this story include the prince's efforts to win through the forest of thorns, the sober view of evil as that which twists what is good into what is misshapen, and the necessary concept of good turning even the deeds of evil people (and fairies) for the better. "Sleeping Beauty" has some wonderful images of love (even beyond the romantic kind) conquering all.
*All images found via Pinterest.