How can I be sure the melons I purchase are ripe (or nearly ripe) and of superior quality?

The best way to judge ripeness is simply to press on the end opposite the stem. If the melon is ripe, it will yield noticeably to the pressure of your finger. Avoid melons with lumps, bruises, or soft spots. Check the melon's navel, or stem end, for excessive softness or mold.
 
Choose cantaloupe that are heavy for their size, with a fruity aroma and thick, well-raised netting.
 
The rind of ripe honeydew melon has a velvet-like texture and a yellow-white to creamy- white color. Ripe honeydew also carries a fruity scent.
 
Watermelons should be free of cracks or soft spots, and the rind shouldn't be too shiny or dull. The bottom should be yellowish in color rather than greenish-white. If the stem is attached, it should look dry and brown. A green stem indicates the watermelon hasn't ripened.
 
Pre-cut melons eliminate some of the guesswork: Look for firm, colorful flesh and dark seeds. White seeds may be a sign of immaturity.
 
Source(s):
Wholefoodsmarket.com