First, the 22 normal fairies:
00 = Fire, red spiky hair, Adventure A3
01 = Water, blue long hair, Adventure P7
02 = Lightning, blonde twintails, Adventure F3
03 = Light, blonde short hair, Adventure H7
04 = Dark, purple braided hair, Adventure N1
05 = Fire, red ponytail, Adventure N6
06 = Water, blue short hair, Great Sea J4
07 = Lightning, blonde short hair, Great Sea B6
08 = Light, blonde long hair, Great Sea D2
09 = Dark, purple short hair, Great Sea A1
0A = Fire, white spiky hair, Master Quest N5
0B = Water, green long hair, Master Quest K8
0C = Lightning, purple twintails, Master Quest B4
0D = Light, brown short hair, Master Quest L2
0E = Dark, white braided hair, Master Quest F5
0F = Fire, white ponytail, Termina K7
10 = Water, green short hair, Termina E4
11 = Lightning, purple short hair, Termina I3
12 = Light, brown long hair, Twilight G1
13 = Dark, white short hair, Twilight D7
14 = Light, pink hair, Koholint Island G6 (DLC)
15 = Water, green hair dark skin, Grand Travels H8 (DLC)
16 = Dark, blonde long hair, Adventure Map L8 (1.6.0)
Most of the remaining values are fully-functional duplicates of the first 22 fairies. A few of them are actually used in the normal game; 6E, 66, and 83 (duplicates of 01, 05, and 0D) belong to the fairies Lazuli, Scarlet, and Magnolia that were distributed through SpotPass.
Values:
00: 17-64, 8C-C7, DB, CD, CE, D4, E9, F0, F5, F7, FA, FC, FE
01: 6E(Lazuli), 72-77
02: 78, 7C-81
03: 86-8B, 82, E2
04: None
05: 66(Scarlet)
06: 70
07: 7A
08: 84
09: None
0A: 65, F1, F8, FF
0B: 6F, D3, D1, D8, DA, DF, E1, E6, E8, ED, EF, F4, F6, FB, FD
0C: 79, D5
0D: 83(Magnolia)
0E: None
0F: 67, C8, E5
10: 71
11: 7B
12: 85
13: None
14: CF
15: None
16: None
These 8 values CRASH the game as soon as the fairy appears on-screen: C9, CA, CC, D7, DC, E0, EC, F2
The remaining 15 values are "glitched". I'm not sure if "glitch" is the right word when you're modifying data like this, but they don't behave like normal fairies.
HYBRID GLITCH: These values give you a fairy with the Fire element and eyes, but with the green short hair of water fairy 10. In all other respects, this fairy seems to function normally. Preliminary testing was not able to cause a crash, so this unique Fire fairy is tentatively STABLE.
Values: D0, D2, E7
DUPLICATE GLITCH: These values create a fairy with no hair, and what appears to be a second body above and behind the first. But bizarre appearance aside, this fairy seems to function normally. Preliminary testing was not able to cause a crash, so these values are tentatively STABLE.
Values: DE (Fire), EE (Fire), F3, (Lightning)
TEXTURE GLITCH: These values create a Dark fairy with a large rectangular body made up of fragments of textures for other things, including fairy foods, eyes, wings, and clothes. The texture changes each time the fairy is viewed.
During battle, the fairy magic button is the fairies' Rental Skill icon.
If you have enough magic to use it, the fairy magic button causes the normal pose and camera angles, but nothing else happens, and the magic meter does not decrease.
These fairies will never speak except when giving you materials or food.
Preliminary testing was not able to cause a crash, so these values are tentatively STABLE, albeit useless because their magic doesn't work.
Values: D9, EA
BLANK GLITCH: These values create an invisible fairy. Not even the clothes are displayed. The stats are also invisible, but they still function.
During battle, the fairy magic button is displayed incorrectly. Each ID uses a different icon, of which two appear to be finished beta elements.
Like with the Texture Glitch, fairy magic plays the normal animation, but the attack doesn't happen, and the magic meter doesn't decrease. This is true for 6 of the 7 values; CB has working Water magic.
Unlike with the Texture Glitch, these fairies speak normally.
Putting clothes on the causes different behavior for different values; 5 of the 7 values CRASH the game, CB doesn't change the invisible appearance, and E3 causes an appearance akin to the Texture Glitch until exiting the My Fairy menu, when it turns invisible again.
The game CRASHES if you're on the glitched fairy and move to another fairy, both in the My Fairy menu and the fairy selection menu before a battle, so these values should be considered UNSTABLE, including CB.
Values:
CB = Water, normal Water fairy button
D6 = Dark, bar 1/3 the size of a normal button
DD = Dark, normal-sized brown smudge
E3 = Dark, interesting hourglass-shaped button (Beta?)
E4 = Dark, semi-transparent speech bubble with tattered edges (Beta?)
EB = Dark, normal-sized distorted purple/white bar
F9 = Fire, tiny purple/white right triangle
EDIT: The next value is only ever 00 in normal gameplay, but increasing it actually changes the fairy's type as well. This increases the number of possible values from 256 to 65536. Obviously, I can't check all those possibilities by hand. I did run through a few, but only found more duplicates and Blank Glitches. The one interesting one was 17 01, which is the pink fairy from Koholint, but of the Fire type instead of Light.