The Best American Short Stories (BASS) is a yearly anthology that is part of The Best American Series published by Houghton Mifflin Harcourt. Since 1915, Best American Short Stories has anthologized more than 2,000 short stories, including works by some of the most famous writers in contemporary American literature. Along with the O. Henry Awards, Best American Short Stories is one of the two "best-known annual anthologies of short fiction." (This summary is from Wikipedia.)
These books are free to read. Each is in the public domain in the United States; check your own country's rules before you copy or share one.