The best way to win writing contests is to write better than everybody else, but there are a few additional tips I can offer.
First, look for contests with narrow qualifying criteria. These can be geographic--for example, in the U.S. there are quite a few contests open only to the residents of individual states. Sometimes there are age limits as well. Here's an example:
The Next Chapter Award
will support an emerging Scottish writer aged over 40 who is yet to publish a full-length work. The selected writer will be offered nine months of mentoring to be arranged by Scottish Book Trust, four weeks on retreat at Moniack Mhor and a £1000 bursary. Applications close 16 May.
The Rhys Davies Short Story Competition
features a brand new Under 21 Prize, and an increased total prize fund of £5000 now on offer. Both competitions are open to all writers born in or currently living in Wales. Entries close 16 May.
By way of contrast, here's one with almost no limits :
The Raymond Carver Short Story Contest is open to writers from around the world and is known to award top prizes to virtual unknowns and never-been-published writers.There's a first prize of $1500 for stories under 6000 words. The deadline for entries is May 15.
This may well be worth entering, but as they take entries from around the world and there are no age or other restrictions, most likely you'd be up against a large number of competitors.
2. Ask how many entries they usually have. Not many contests list this in their promotional material but some of them will tell you if you ask.
3. Take the entry fee into account. A high fee limits the number of entries especially if the money prize is low. Normally that's a good reason to avoid entering, but if there is another benefit (e.g. a residency, meetings with agents, etc.) and you feel you have a strong entry it may be worth going for it.
4. Read and follow the rules carefully. This sounds obvious but you would be surprised how many people don't adhere to the required word count, for example.
5. Consider contests which require entries on a particular theme or mentioning a location, etc., like this:
Tell It Strange Essay & Story Contest
is for entries up to 1000 words written in response to one of three selected Annie Proulx quotes. First prize is US$1000, publication in The Writer magazine and a creative writing workshop in New York City or online by Gotham Writers Workshop. Entries close 31 May.
For more generic contests, people will dig up something they've already written. Responding to a more specific topic or style usually means sitting down and writing something new, and fewer people will put in that effort.