Lots of good tips by others here. Of course, it is hard to find certain fruits at peak ripeness at a supermarket, usually because the peak ripeness window is so short and the fruit is so fragile at that time.

Two of my favorites, to ripen at home:

Peaches, and this works for nectarines as well.
There is nothing better than a fragrant, juicy peach with skin that will pull off effortlessly. But a tender peach does not keep long or well, especially in a bin with 1,000 others. So, how to pick when they are all rock hard? Make sure that the peach, except for where it blushes red, is yellow. Some peaches are almost entirely red, so look at the stem end. Avoid peaches that are green, rubbery, wrinkled, or bruised.

If you choose a peach that has a base color of yellow, it will be as sweet as possible. Ripening at home changes the texture of the fruit, but not the sugar content, and a yellow fruit was on the tree longer than a green one. If you are impatient, a crisp peach is also good sliced into salads, added to sangria, and basically is an interesting substitute for raw apples in many places.

I adore a good mango. If you are lucky, you will find a ripe, or near ripe fruit at the store. It will yield slightly and without bruising when gently squeezed. The entire fruit will smell sweet and strongly of mango; stick your nose right up to it in several places. The skin will be cool and smooth. There is true tactile pleasure to be had in caressing a ripe mango. Check the stem end. If the nub of the stem falls off at the least provocation, it is beginning to be overripe. If the mango is turning light brown around the stem, with the color radiating outwards, that part has become overripe and the entire fruit becomes suspect. If the stem end is wrinkled, it is another sign that the mango is starting to head downhill.

Unfortunately, most of the ones in supermarkets are either attractive rocks or scary, yellowing near-raisins. I avoid the near-raisins. They may be sweet and delicious, but I've never had the courage to try one. Instead, I purchase firm ones and ripen them at home. Ahh, the impatient peeping into the bag every evening, the judicious gentle pinch, and the questioning sniff.... Ahem. Where was I? Oh yes, it is easy to pick a good candidate for home ripening. Don't go by color. Instead, smell it. As I said before, stick your nose right up to it and sniff in several places. If the fruit is approaching ripeness, the stem end will smell sweet. That isn't enough. If the opposite end smells pleasantly fruity and somewhat sweet, that mango is a keeper. It will ripen in a reasonable amount of time, and it will be sweet and juicy. If all the mangoes just smell like the box, well they will take a very long time to ripen, and may dry out before they ever get soft enough to eat. As always, try to avoid bruised, wrinkled, or otherwise depressed fruit.

As previously mentioned, use a paper bag to ripen peaches or mangoes. Place the fruit in one layer in a clean brown paper bag. This is particularly important if you are storing a large quantity of fruit, as the fruit's own weight can bruise individual pieces as they soften.

I like to place the bag on its side so I can fit in more fruit. I try to match the quantity of fruit to the size of the bag so there isn't too much dead space in the bag. Fold the open end of the bag closed up to the fruit, and keep it at room temperature or just a bit warmer. Add an apple if you want to accelerate the process. Check every day as some will ripen sooner than others. There is nothing sadder, after all that waiting, than a perfect peach going moldy or a mango turning brown.