The horn is a single spiral horn in the middle of the forehead.
They have a palpable aura of magic, holiness and wildness that quite overwhelms any sense of ridiculousness at their gangling builds and those jolly unwieldy-looking horns. They never, ever 'feel' tame, even if caged.
Unicorns are never seen dirty or dishevelled unless captured or killed.
Unicorns are solitary with territories generally encompassing a forest. A resident unicorn is what enables other magical creatures (hobgoblins, boggarts, brownies, sprites etc - it hasn't yet been decided which of these are found on Larkspur) to live in the surrounding area.
There is a tradition, stretching back into legend and continuing into present times, of knights keeping themselves pure, going and finding a unicorn, and riding it on a quest. Generally when the monster is slain and the maiden rescued, the unicorn makes a hasty exit.
Unicorns are connected with purity in other ways: for example, they can purify water by dipping their horns into it, and indeed always do this before they drink.
An imprisoned unicorn's aura changes to suggest a feeling of outrage and sacrilege to any onlooker.
Unicorns can be killed if you want to incur a godly curse on yourself and your entire bloodline. Once killed, they are usually dead.
The horn of a dead unicorn has no special properties whatsoever. A chip from the horn of a live unicorn has very many special properties indeed. Just watch out that the unicorn doesn't die while you're using your unicorn-ivory-tipped wand.