I know this is bending the rules, but I've seen single horned quadrupeds, a half a dozen at least in my life. Some more convincing than others, but none with wings...
That'll be a tricky one, I'll bet.
My point is that a 'version' of unicorns have and do exist, if not as they has been fictionalized, but they do exist in some form.
Have you ever heard of the lumberjack who work alone with a black ox who glistened blue in the sun, he hefted a hand made axe, and he and his animal could do the work of 2 men and two horses. He stood a half a man taller than most, and was an axe handle wide at the shoulders.
You likely know him as the inflated Paul Bunyan. Some claim there was a real man behind the myth...