Store fresh asparagus in the refrigerator. To keep them crisp, trim the woody ends and place the asparagus upright in a glass jar with about one inch of water at the bottom. Cover the tops of the asparagus loosely with a plastic bag and store them in the refrigerator for up to one week. You can also wrap the spears in a damp paper towel and place them in a plastic bag, but they won't last for quite as long this way.