Most herbs you can store in the fridge, but the exception is basil. It's susceptible to the cold, so just snip of the ends and put it in a glass of water, like a bunch of flowers. I sometimes chop up parsley and freeze it - it keeps for ages that way.
One thing to remember with herbs is not to wash them until you need them, and wrap them loosely. Both of these will reduce the chance of mould.
It is best to use the herbs as soon as possible, but if you don't plan on using them straight away, the best way to store them is to first wash the herbs, then dry them, and seal them in an air tight container such as a zip bag of clip box. You should then put them in the fridge and use within a week.
Put in a mushroom paper bag as loosely as possible to keep as dry as possible.. then into a plastic bag to avoid moist from surroundings. I avoid washing until I will use the herbs as there are bacteria in water and moisture helps bacteria to grow..