Chicken drumsticks are no different from other types of chicken in terms of how long you can store them in the refrigerator. All chicken cuts only stay fresh in your fridge for about two days. If you don't plan on using your chicken drumsticks on the same day, or at least by the day after you buy them, put them in the freezer. If you're defrosting your chicken in the fridge, it will be good for about two days from the time it's fully defrosted. If you do put the drumsticks in the freezer, make sure they're in an airtight container.