I guess it's a learned trait, but I thump the melons with my middle finger  until I find one that sounds just right!!       I can't describe it because I think it's more psychological than anything, which is what most of you were expecting a psycho like me to say! 

I spoke to a watermelon buyer (wholesale) recently, he said there isn't a good way to judge them by looking, but he told me if I got a bad one I should return it.  He said the grocers will always be glad to replace them.

I haven't had to check out his advice yet this year, but I love watermelon and I really can't stand grainy, over-ripe ones so it's just a matter of time.

Watermelons should feel heavy and sound hollow when thumped.

For other types of melons use your nose. Look for strong fragrance and sweetness.