One of the greatest openings ever is to a short story - but it's only great BECAUSE it's a short story. The Secret Life of Walter Mitty. I'd have to re-read it to remember which event it opens on but you're kind of left not knowing if it really happened, or if it's all in his head - or if there's another level of reality altogether.
The best opens engage the reader somehow - either by tossing them into the action, by introducing them to a situation that they should want to know more about, by letting them see a hint of a mystery.
Some all-time classic opens:
"In a hole in the ground, there lived a hobbit."
"It is a truth universally acknowledged, that a single man in possession of a good fortune, must be in want of a wife."
"It was the best of times, it was the worst of times, it was the age of wisdom, it was the age of foolishness, it was the epoch of belief, it was the epoch of incredulity, it was the season of Light, it was the season of Darkness, it was the spring of hope, it was the winter of despair, we had everything before us, we were all going direct.to Heaven, we were all going direct the other way.
"It was a bright cold day in April, and the clocks were striking thirteen."
"The Miss Lonelyhearts of the New York Post-Dispatch (Are you in trouble? Do-you-need-advice? Write-to-Miss-Lonelyhearts-and-she-will-help-you) sat at his desk and stared at a piece of white cardboard."
" You don’t know about me without you have read a book by the name of The Adventures of Tom Sawyer; but that ain’t no matter. That book was made by Mr. Mark Twain, and he told the truth, mainly."
"One morning, as Gregor Samsa awoke from anxious dreams, he discovered that during the night he had been transformed into a monstrous bug."
"Someone must have slandered Josef K., for one morning, without having done anything truly wrong, he was arrested."
"You will rejoice to hear that no disaster has accompanied the commencement of an enterprise which you have regarded with such evil forebodings."
" Once, there were four children whose names were Peter, Susan, Edmund, and Lucy. This story is about something that happened to them when they were sent away from London during the war because of the air-raids."
"All this happened, more or less."
"Mr. & Mrs. Dursley, of number four, Privet Drive, were proud to say that they were perfectly normal, thank you very much."
"The man in black fled across the desert and the gunslinger followed."
"Far out in the uncharted backwaters of the unfashionable end of the Western Spiral arm of the Galaxy lies a small unregarded yellow sun."
" It was love at first sight. The first time Yossarian saw the chaplain, he fell madly in love with him."
" Alice was beginning to get very tired of sitting by her sister on the bank, and of having nothing to do: once or twice she had peeped into the book her sister was reading, but it had no pictures or conversations in it, “and what is the use of a book,” thought Alice, “without pictures or conversations."
(this list comes from a site claiming to have the 100 best opening lines or paragraphs in all of fiction; this is the only one I felt I had to include, even though I never read the book, because it kind of fits): "This is my favorite book in all the world, though I have never read it."
"Stately, plump Buck Mulligan came from the stairhead, bearing a bowl of lather on which a mirror and a razor lay crossed."
" No live organism can continue for long to exist sanely under conditions of absolute reality walked alone."
"The story so far: In the beginning, the Universe was created."
" Sam Vimes sighed when he heard the scream, but he finished shaving before he did anything about it."
"Somewhere in la Mancha, in a place whose name I do not care to remember, a gentleman lived not long ago, one of those who has a lance and ancient shield on a shelf and keeps a skinny nag and a greyhound for racing."
"That was me, that is Alex, and my three droogs, that is Pete, Georgie, and Dim, Dim being really dim, and we sat in the Korova Milkbar making up our rassoodocks what to do with the evening, a flip dark chill winter bastard though dry."