You can start a sentence with "however," but you need to be careful. And what about "and" and "but"?
Starting a Sentence with "But" Instead of "However"
And here’s something that may surprise you even more: modern sources such as The Chicago Manual of Style, Garner’s Modern American Usage, and others say that although it isn’t wrong to start a sentence with however (9, 10, 11), it’s usually better to start a sentence with but. Yes, many of you were probably also taught that it’s wrong to start a sentence with a conjunction such as and and but, but that’s a myth too.
After saying it’s not an error to start a sentence with however, Chicago goes on to add “however is more ponderous and has less impact than the simple but,” and Garner’s sentiment is also that it is more effective to start a sentence with but or yet than however.
They would probably prefer that Spock told Bones, “I intend to assist in the effort to reestablish communication with Starfleet. But if crew morale is better served by my roaming the halls weeping, I will gladly defer to your medical expertise.” On the other hand, you also have to consider the style, and you could argue that however is a better fit for a character such as Spock precisely because it sounds more ponderous.
When to Avoid Starting a Sentence with "However"
Even though it's not wrong to start a sentence with however, sometimes it’s still a good idea to avoid it because a lot of people think it's wrong. I don’t advise starting a sentence with however in a cover letter for a job application, for example. You don’t want your resume to get dumped because someone thinks you’ve made a mistake when you really haven’t.