Yes: in fiction.
No: in formal/academic papers, or in many non-fiction books.
Thomas Pynchon is one of the greatest American novelists and a great stylist. You'll find in his work:
"Bigger, not softer, ése. And speaking of lunch, how about
tomorrow at Vineland Lanes?"
Zoyd had played a few mob weddings in his career, nothing the kid
couldn't handle, and besides the eats would more than make up for any
awkward episodes, so it wasn't as if he were running a mean trick on
his daughter's boyfriend, whom he was still not 100 percent crazy
about, or anything like that. And as a problem to be addressed,
Isaiah was more like a vacation from deeper difficulties, chief among
which, all of a sudden, was the recrudescence of Hector Zuñiga in
Zoyd's life, a topic, as he lit a joint and settled in front of the
soundless Tube, that his thoughts unavoidably found their way back to.
Vineland, by Thomas Pynchon
A Grammar Book for You and I-- Oops, Me!: All the Grammar ... C.
Edward Good - 2002
When you exercise your new writing muscles and use conjunctions to start sentences, make certain you do not put a comma immediately after
the section that starts with
Never Start a Sequence with a Conjunction
Poppycock, on p. 157
which includes comments from the great stylists of the past, such as Fowler and Follett.
The most exhaustive essay on the matter that I know is:
Garner on Language and Writing: Selected Essays and Speeches of Bryan A. Garner By Bryan A. Garner. See:
On Conjunctions as Sequence Starters at p. 63
Garner being the preeminent style expert of our time, IMO.
I'd strongly suggest to at least peruse it.