Which is the correct spelling: “fairy” or “faerie”?

As others have noted, fairy is the standard modern spelling, and faerie is a pseudo-archaism.

However, in some contexts there is now a semantic distinction between the two spellings! In particular, fairy tales and the associated idea of fairies typically refer to the genre of folk stories printed by the Brothers Grimm, then sweetened and popularized for modern audiences by Disney et al. Faerie stories, on the other hand, are stories about the fae: otherworldly, unpredictable, and dangerous creatures who appear in the folk-tales and myths of England and Ireland. In origin, of course, the fairies and the fae are one and the same, but the connotations and usage of the words today are headed in opposite directions.

A Google image search for fairy vs. faerie shows a lot of overlap, but some very significant differences in tone and content between the two terms. Note that the search for "fairy" turns up a number of simple cartoon drawings and classical, idealized figures, while the search for "faerie" has a higher proportion of non-human and sexualized figures.

Here we have a difference between sources: the New Oxford American Dictionary reports faerie (and faery) as “late 16th century (introduced by Spenser): pseudoarchaic variant of fairy”. On the other hand, etymonline has “ supernatural kingdom, "Elfland," by late 14th century, from Old French”.

At least one available data point weighs, to me, in favour of etymonline (and thus against NOAD): Oberon, the faery prince: A masqve of prince Henries was first published in 1616.

Going further, I looked at the French féerie, which derives from Old French faerie (circa 1188), itself from fae (modern writing: fée) (1140), itself from Latin fata (deities of destiny, or Parcae). This is unlike fate itself, which comes from Latin fatum, -a, (“oracle, prediction”).

So, fairy, fay, faeries, farye and the Fates comes from the same Latin root fata, while fate itself comes from the different fatum, -a. Blows my mind!

I would say the former is correct in modern usage, a descendant of the latter which is seemingly from Old French / Middle English.

From The free dictionary:

For 'Fairy'

[Middle English fairie, fairyland, enchanted being, from Old French faerie, from fae, fairy, from Vulgar Latin Fta, goddess of fate, from Latin ftum, fate; see fate.]

For 'Faerie':

[Middle English faierie, fairie; see fairy.]