Look at its belly: Watermelons have an underside, or belly, which is in contact with the ground throughout its growth, called a ‘field spot’. This spot on a ripe watermelon will be yellowish (sometimes referred to as “buttery”), and not white, which indicates an unripe melon.
Thump it: Using your knuckles, rap on the middle of the watermelon while holding it up to your ear, or flick it with your finger (like flicking a crumb off your shirt). A ripe melon will have a hollow sound when knocked, which sounds more like a ‘plunk’ than a ‘thwack’. An unripe watermelon will have more of a higher pitched sound, while an overripe one will make a ‘thud’ or a lower-pitched sound. Learning the difference between the sounds of an unripe vs. a ripe watermelon takes a bit of practice, but one way to get a head start on it is to ask a melon grower at your local farmers market (or perhaps the produce manager at the grocery store) to demonstrate it for you so you can hear it for yourself.
Sniff it: Pick up the watermelon and carry it a little bit away from the rest of the melons (so you don’t pick up the smell of other fruits), and give it a good sniff. A ripe watermelon should smell slightly sweet, and similar to what fruit tastes like, but not overly sweet (which can indicate an overripe watermelon). This sniff test also works excellent (actually, even better than for watermelons) on other types of melons, such as cantaloupes and honeydew.
Squeeze it: Gently squeeze the side of the watermelon to see if there’s a bit of ‘give’ to it. The rind of the melon shouldn’t be soft, as the skin of some fruits get when ripe, but it also shouldn’t be hard as a rock, with no give to it at all.
Heft it: If you’ve gone through the above tests and still can’t narrow down your choices between a couple of watermelons, try comparing the weight of similarly-sized ones and pick the one that seems heavier to you. This isn’t a failproof test, but I’ve found it to be reasonably reliable (although since I don’t tend to buy two similar melons and compare them after cutting them open, this may be more of an indication that I’ve already picked the ripest ones with the thump test).