Bunches of grapes can easily harbor mold-friendly moisture in their crevices, so it's important to store them dry and unwashed. Grapes can be stored safely for weeks in the refrigerator or, once cleaned and dried well, in the freezer for a longer period of time.
According to Real Simple, the best place to store grapes is the refrigerator, in a resealable plastic bag. If possible, use a breathable plastic bag. Grapes can ferment and grow moldy when stored at room temperature, so it's best to store them in a cool environment. Keep grapes stored unwashed until you are ready to use them. They can be stored for about two weeks in the refrigerator, or up to three months in the freezer.