I rarely recommend writing contest that charges a reading fee. In general, if there is a fee involved it is best to avoid it like the plague. There are rare exceptions. Most contests I mention have no fee.
Glimmer Train – For literary fiction. One of the rare exceptions, though they do have some non-fee contests too. This is a very prestigious contest. If you’re very polished, and not doing genre fiction, enter this one.
Mattia Family International Poetry Contest – No entry fee and cash prizes. Nice folks too. This is obviously a labor of love.
Writers of the Future – If you are very polished and a genre writer, this is the contest for you, at least if that genre is science fiction, fantasy, or horror. There is No entry fee, it is very prestigious, and the winners receive substantial cash awards. It is open four times a year. They also have a contest for illustrators.
Your Story – A nice no fee contest for short fiction, run by Writer’s Digest. It is open roughly every other month. Wining stories are published in the magazine.