The short answer is in a bag in the refrigerator and use them quickly.
The long answer is that if you don't have a cat or culinarily curious sort of pet, you can make your herbs last longer and have them double as a sort of cheap bouquet. All you do is fill a short glass with some water, take off any nasty brown or yellow leaves, and plunk the herbs in your "vase." Set the glass on your table in some sunlight. If the water gets cloudy change it for some cleaner water. Don't let leaves get into the water.
This method works beautifully for parsley, mint, and I think basil but NOT cilantro. As I learned to my sorrow (so frequently that I wonder now that I didn't realize this at once), cilantro must stay dry in a sealed plastic bag in the refrigerator. The exception to this rule is that you can first de-stem the cilantro (and save the stems for stock or cooking dried beans), wash the leaves, dry them, and then reseal them in a Ziploc bag.
Butternut Squash muffins will save your soul - The MOST forgiving muffin recipe ever (as in, you can make a zillion substitutions and they will still taste really, really good) is from Vegan With a Ve...