The Bath Short Story Award, the Bristol Short Story competition, the Bridport Prize, and the BBC National Short Story Award all have associated anthologies containing their short-listed stories. Below are some non-competition prose anthologies that you might aim for -
- Best British Short Stories (Salt) - Yearly. Chosen from published stories (WWW and paper).
- Unthology (Unthank books) - Twice a year. See the submission guidelines at http://www.unthankbooks.com/unthology.html.
- Comma Press do various collections - sometimes commissioned. Their submissions page is a useful read even if you don't plan to submit to them.
- the Siren are beginning to publish anthologies - see their submissions page.
- Riptide do various collections, more often themed nowadays. See their submissions page.
- The Fiction Desk publish quarterly collections - sometimes themed. There's a voluntary submission fee of 2 pounds. See http://www.thefictiondesk.com/submissions/short-story-submission-guidelines.php.
Magazines like the monthly Writing Magazine have calls for submissions to less regular anthologies - usually themed. Vanessa Gebbie pointed out that there's an open call for story submissions for an anthology to be published next Autumn by Freight books, on the hundredth anniversary of Einstein publishing his Theory of General Relativity - see the Call for entries (closing date 28 Feb, 2015)
A word of warning - in the poetry world people sometimes invite submissions for an anthology, printing most of what they receive and expecting contributors to buy a copy. Prose is far less prone to these money-making schemes, but it's worth sticking to the established publishers if possible. And of course it's a good idea to buy their books.