Glimmertrain is a short story market in the US which offers a variety of competitions throughout the year. They also have certain months when they accept Standard Submissions, which are free to enter. The other competitions have an entry fee, but the reward is much higher prize money.
Submissions are made online and, once contributors register, you can keep a check on whether or not your story has been read. It seems to be a very worthwhile publication, although I have yet to order a copy to do proper market research of the type of stories they have published!
Full submission details are on their excellent website.
Why not give it a try.