When I lived in Hong Kong, I had a good friend who taught me her method of picking the ripest, best-tasting melons. She came from Thailand, land of many fruits, and she taught me to rap all of the melons in the pile and listen for the deepest tone.
While there may be some validity to this method, I recently read an article that indicated that the key thing to look for is a creamy spot on one side. This spot indicates that the melon was allowed to rest in one spot in the field over a long period of time, allowing ample time to gain natural sugars and water content, making for the best flavor!
As for shape, avoid asymmetrical melons, because this indicates that the melon may have been water starved at some point and grew in irregular spurts.
Another interesting suggestion was to avoid refrigerating the melon, as this will alter it’s true flavor. One farmer even puts his watermelon in the hot sun and slices them up hot, claiming that the heat brings out all of the fruit’s exotic, subtle nuances in flavor, describing it as "mango-like". Now that I have to try!