I love herbs. Every time I buy a bunch of fresh herbs, I baby the bunch until I either use it up, or it goes bad. After many different experiments, I have found out that the best way to store the herb bunches is as follows:
Cut the bottoms of the stems of the herb bunch.
Put the herbs in a container with about an inch of water.
Next, cover the herbs with a plastic bag.
Store in the fridge and change the water if it gets cloudy.