According to a usage note in the fourth edition of The American Heritage Dictionary, "But may be used to begin a sentence at all levels of style."
The same point was made over a century ago by Harvard rhetorician Adams Sherman Hill: "Objection is sometimes taken to employment of but or and at the beginning of a sentence; but for this there is much good usage" (The Principles of Rhetoric, 1896). In fact, it has been common practice to begin sentences with a conjunction since at least as far back as the 10th century.
Still, the superstition persists that and and but should be used only to join elements within a sentence, not to join one sentence to another. Here, for instance, is an edict found recently on an English professor's "Composition Cheat Sheet":
Never begin a sentence with a conjunction of any kind, especially one of the FANBOYS (for, and, nor, but, or, yet, so).This same fussbudget, by the way, outlaws the splitting of infinitives--another durable grammar myth.
But at least the professor is in good company. Early in his career, William Shawn, longtime editor of The New Yorker magazine, had a penchant for converting initial buts into howevers. As Ben Yagoda reports in When You Catch an Adjective, Kill It (Broadway Books, 2007), Shawn's habit inspired one of the magazine's writers, St. Clair McKelway, to compose this "impassioned defense" of but:
If you are trying for an effect which comes from having built up a small pile of pleasant possibilities which you then want to push over as quickly as possible, dashing the reader's hopes that he is going to get out of a nasty situation as easily as you have intentionally led him to believe, you have got to use the word "but" and it is usually more effective if you begin the sentence with it. "But love is tricky" means one thing, and "however, love is tricky" means another--or at least gives the reader a different sensation. "However" indicates a philosophical sigh; "but" presents an insuperable obstacle. . . .Of course, like any other word, but may be overworked. To avoid puzzling or distracting our readers, we should probably ration our use of it: say, no more than one but per sentence, and never at the start of consecutive sentences. But these are only rough guidelines, not hard and fast rules.
"But," when used as I used it in these two places, is, as a matter of fact, a wonderful word. In three letters it says a little of "however," and also "be that as it may," and also "here's something you weren't expecting" and a number of other phrases along that line. There is no substitute for it. It is short and ugly and common. But I love it.
In any event, don't start scratching out your buts on my account.