The Griffin and the Minor Canon is a modern fairy tale by Frank R. Stockton, which first appeared in the St. Nicholas Magazine, October 1885. While perhaps not as famous as his The Lady, or the Tiger?, it is likely to be of greater interest to younger readers.
There once was a church with a very nice carving of a griffin. It remained thus for many years, until one day a passing bird saw it, and eventually passed on news of its existence to an elderly griffin who happened to live not too very far away. The griffin, having no mirror nor any other griffins living nearby, had no real idea as to what he might look like, and so decided to find this church and view the carving. Once the griffin arrives in town he causes considerable panic, leaving no-one but the minor canon to deal with him.
The minor canon is not entirely happy with this, but fears if he doesn't deal with the situation the griffin may start eating people. The griffin turns out to be quite friendly, and determines that in addition to frequently admiring the carving, he should also assist the canon in his churchly duties. He is quite good at this, and assures the canon (and through him, the townspeople), that they need not fear him, as he eats only on the equinoxes. As the date of the autumn equinox approaches, the townspeople bully the minor canon into leaving the town in hopes that the griffin will follow... but instead the griffin takes it into his head to assume all of the minor canon's duties.
This is an excellent lighthearted fairy tale (although the ending is not entirely happy). While it was originally intended for children, and is still quite appropriate for the sort of child who is not daunted by vocabulary like 'minor canon', it is also frequently enjoyed by adults. The story is verbose but friendly and humorous, and presents the fantastical beast as a perfectly normal aspect of the world that the townsfolk happen to be unreasonably terrified of. It is essentially Victorian era magical realism.
Over the years, it has appeared in Isaac Asimov's Magical Worlds of Fantasy # 6: Mythical Beasties, Alfred Hitchcock's Tales of the Supernatural and the Fantastic, The Oxford Book of Modern Fairy Tales, Neil Gaiman's Unnatural Creatures, and a number of other collections, including Stockton's own collections, The Lady or the Tiger and Other Stories and The Bee-Man of Orn and Other Fanciful Tales. In 1964 an edition was published as a stand-alone children's book illustrated by Maurice Sendak, and although I have not yet read this edition, those who have tend to remember it fondly. Moreover, it is out of copyright, and available online through Project Gutenberg or Classic Shorts.