Looking for a publisher? Here are some independent publishers that accept submissions of fiction. Note that for most of these publishers, you first submit a query, synopsis, and sample chapters. If they are interested, they will ask for the full manuscript. Some small publishers have started hosting contests with substantial reading fees. The winner of the contest gets their book published. I do not recommend this strategy, as odds of publication are low and the fees add up quickly (usually they are at least $25). The publishers listed below do not have submission fees.
For further research, Poets and Writers has a list of small publishers. Writer’s Market (published yearly) is also a helpful resource.
Black Lawrence Press — free open reading period is June 1-June 30
Coffee House Press — watch for their open reading period
Jellyfish Highway (UPDATE 5/4/18: Defunct)
Leapfrog Press (UPDATE 5/1/17: A commenter notes that Leapfrog now charges $33 for fiction submissions, which they’re only accepting through a contest. This is an unfortunate development. Once again, I don’t think writers should pay to submit to contests, as chances of publication are slim.)
Louisiana State University Press (Yellow Shoe Fiction Series)
Two Dollar Radio ($2.00 fee–I included this since the fee is small)
Vandalia Press (imprint of West Virginia University Press) — reopening for submissions in 2017
Emily Eckart 
Great resource! Thank you for sharing these.
Warning! Leapfrog Press is charging authors a $33 reading fee for fiction submissions through their “contest.” Rip off!
Thanks for the update! I’ll put a note in next to their entry.