My players have demanded some unusual characters. Based on their options, I believe that AD&D is the way to go. Page 21 of the DMG offers some advice for these unusual characters.
I started with the Unicorn Stats from Monster Manual. AC: 2, HD: 4+4, Damage: 1d6/1d6/1d12, +2 to hit dice on a charge, charge damage is 2d12. They are immune to poison, surprise on 1-5, detect opponents out to 24", they save vs. magic as an 11th level Magic User.
The DMG suggest limits. Cool. Limit one: The best unicorns are equal to 11th level magic users. Limit 2: The best unicorns have 4+4 hit dice at max level which equates to 36 hp.
I like those numbers. If I take 36 and divide by 11, that gives me 3.27 which is close the to average of a 1d6. The one issue I have with this is, unicorns are big animals so 1d6 at first level is too small. I'll give them the basic 1d8 for monsters and fighters at level one, but then drop the die to 1d4 for subsequent levels. Without Constitution bonuses, this will max out at 48 hp. This is a third better than the monster version, but since D&D has always had "superior" versions of standard monsters, I can work with this.
AC is tricky. I could do a scale like monks, but decided against it. Unicorns start with AC 2 and keep that number their whole life. To balance this, I will adjust the amount of damage and attacks by level and not allow them to wear barding. At low levels they can't attack with their horns because they have an aversion to touch. As they age, they learn how to use the horn as a weapon. Also, all attacks improve as they progress through levels. Initially, they start with 1 attack per round, but later on they can get in two kicks per round. Major attacks with the horn and the bucking kick are limited to once per round.
Since they are magical and sneaky, they will have the ability to move silently like thieves, cast spells as a natural ability and lay hands like paladins.
I was going to type out everything, but I think a schedule will do.