I can not believe I didn't know about this simple trick years ago! I have only done this on parsley and cilantro, so I can not recommend anything else.
Take your herbs, and place them in a cup with about 3 inches of water, cover with a plastic bag and place in the refrigerator. I've seen a lot of posts about clipping the stems and making sure the leaves are dry, but I have never gone that far. Lazy maybe? No joke, they will last a month and still be fragrant and healthy. Talk about never throwing out molded greens again!