These are all blogs or list sites, so more for those who don't mind sifting through rather than having a searchable database:
Angie's Desk is the blog of writer Angela Penrose. Every month she posts up anthology deadlines - it's also her all purpose blog so filter by using the tags "anthology market listings" or "open submissions".
Publishing ...and Other Forms of Insanity is the blog of Erica Verrillo, who posts publishing news, open submissions, and contests.
Cathy's Comps and Calls specialises in free to enter calls that can be entered online.
Some websites that have a section of contest and competitions are:
Trish Hopkinson's website
Christopher Fielden's website
Almond Press
Creative Writing Ink
Writers Reign
Mslexia
Writing Magazine (this only their own competitions)
Obviously as these aren't searchable they may also list markets that don't pay or charge fees. As always, read the guidelines before submission. Happy hunting!