I don't believe there is a fool-proof way to determine 'ripeness' without taking a slice out of it.
The best you can do is look for certain signs: Ripe melons have a hollow sound when you tap or slap the outside Look for the patch where the melon would have been on the ground (called the field spot).
If it's a yellow colour its probably ripe, if it's white, it's probably not.
It should feel relatively heavy when lifted Weird areas on the skin aren't necessarily bad.
insects may have tried to start eating the fruit because it is ripe, but have only marred the surface.
Unfortunately, melons don't continue to ripen once picked, unlike fruits such as apples, bananas etc.
which contain ethylene.
As a tip don't store melons with these kinds of fruit, they may well go 'soggy'.
Extra:
NYtimes video on picking the right watermelon .
cooking.stackexchange.com/users/115/Pulse
As pulse said, colour is a good indicator and give them a tap
and they'll have a nice hollow sound.
The other thing I do is pick them up...
I don't know why, but ripe melons tend to feel "heavy" for their size.
cooking.stackexchange.com/users/125/lomaxx
Shake it.
If you hear things moving inside, it's overly ripe.
Press it.
If it squeaks a little, it's just OK.
If it doesn't, it's either unripe or already wilting.
Knock on it with your finger, like if you'd be knocking on a door.
It has to sound just a tiny little bit hollow.
Too hollow means overly ripe, not hollow at all means unripe.
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For melons other than watermelon, always smell them, they should have a good aroma.
Unfortunately this does not usually work for watermelons.
The color and weight are usually the best indicators.
cooking.stackexchange.com/users/51/Janelle
I just saw a post on rulesofthumb.com that says: A watermelon is ripe when you hear "punk" rather than "pank" or "pink" when you tap it with your finger.
cooking.stackexchange.com/users/210/Sam Holder
I have found that if you look on the bottom of the watermelon, (where it sits on the ground), if it is yellow and the lines are straight and yellowish green then the watermelon is sweet and ripe.
If the lines are a light green and wavy then the melon is not sweet or ripe.
cooking.stackexchange.com/users/25596/Connie
Knock on it lightly.
If someone answers, you've got a very special melon!
Actually, if it sounds somewhat hollow, it is ripe.
As a child we would "plug" the melons by cutting a 2" x 2" trianglebout of it.
That's a fool proof way to see if your melon it ripe.
cooking.stackexchange.com/users/35707/Ric Sabatini
For a sweet melon it normally takes a high sulfer soil for a sweet melon.
So you may not find any sweet ones to buy.
Thump or knock on the melon it should sound hollow.
Melons can not be picked ripe.
They bust to easy to ship in totes.
So set them in the sun for a few days.
Once you wax them to hold in moisture.
cooking.stackexchange.com/users/54532/J Bergen
Lately, every watermelon I bring home form the organic section of the supermarket has not been sweet.  Among the small, seedless varieties, I try to pick the densest. How do I know if it is ripe?  Can I ripen them at home?  Should I keep them in the fridge?
How do I pick a watermelon at the supermarket?
cooking.stackexchange.com
<ripe><fruit><watermelon>
cooking.stackexchange.com/questions/914
cooking.stackexchange.com/users/183/papin
While it is difficult to know for sure without opening the watermelon, the patch where the melon was resting on the ground should be yellow with straight and yellowish-green lines. When you tap the watermelon, it should have a hollow sound. For other kinds of melons, a sweet smell is a good indicator of ripeness. While there isn't a full-proof way of telling that a melon is ripe, you can look for a good aroma, color and weight. Try tapping it to see if it sounds hollow.
There is non full-proof way to tell that a melon is ripe. Try tapping it to see if it sounds hollow. This is a sign it is very ripe. Color and weight are good signs of ripeness. The melon should have a good aroma.