There are loads of sites online where you can read the entire texts of fairy tales or fairy tale collections for free. These are all that I've found so far.
First, folklore or world tales:
Tales of Wonder is an archive with a sampling of folk and fairy tales from around the world, including Central Asia, Central Europe, Scandinavia, Japan, Africa, the Middle East, and more.
The InfoPoland site at SUNY Buffalo has a collection of twelve Polish fairy tales.
Blackmask Online has an etext of Japanese Fairy Tales.
The Armenian Embassy has a collection of sixteen Armenian fairy tales.
Italiansrus.com has a page of Italian fairy tales.
Next, collections by specific authors:
An online collection of more than 656 of Aesop's Fables at Aesopfables.com. Translations are either Rev. George Fyler Townsend (1814-1900), Ambrose Bierce (1842-1914), or Jean de la Fontaine's French rendered into English by "several good internet souls."
National Geographic has a nice, only mildly cheesy presentation of Grimms' Fairy Tales: from Folklore to Forever, twelve tales based on a 1914 translation.
Grimm's Fairy Tales [sic] and Tales Collected by the Brothers Grimm are two scholarly collections of tales, both based on the 1884 translation by Margaret Hunt. The maintainer of the 209 Grimm's Fairy Tales recommends the other collection, which is more accurate. Tales Collected by the Brothers Grimm includes 200 tales and ten legends, which are all available individually or as a single zip file.
All of the preceeding links are via The Daily Pick. Now for some of my own.
Mythfolklore.net has a large collection of tales from Andrew Lang's fairy books.
With the Proppian Fairy Tale Generator, you can select the mythic functions you want, and from them generate a semi-coherent tale based on Vladimir Propp's model. I would have included this in Make-Your-Own, if I'd remembered.
Via The Daily Pick.
"Wakey, wakey: Sleeping Beauty hasn't always been a docile object, says Lyn Gardner. She's had to face a cannibal queen, a rapist king - and even a Nazi prince." An oldish but still interesting article in The Guardian from December discusses the history of Sleeping Beauty and the changes she's undergone through time.
SurLaLune is simply the best resource out there for everything fairy-tale related. There are articles, illustrations, book recommendations, annotated tales, full texts of books, a discussion board, and lots and lots of links. Just about all you could need.
The Endicott Studio is another great all-around resource for fairy tales, folklore, and myth. Home to the Journal of Mythic Arts, they have a reading room of excellent articles and fiction, a poetry coffeehouse, an art gallery, multimedia arts, a discussion board, and more.
I am tempted to post a few more links dealing with myth right now, but it does say "fairy tales" in the header, so I think I'll save all that for another time.
UPDATE: I forgot one of the links I really wanted to post. McSweeny's offers up a great list of Klingon Fairy Tales, including such classics as "Goldilocks Dies With Honor at the Hands of the Three Bears" and "There Was an Old Woman Who Lived in a Shoe With a Big Spike on It."