Asparagus like any other type of food can go bad if it is left out long enough so you will want to make sure to store it properly so you will not have to worry about this for a while after you bring it home from the supermarket. In the refrigerator asparagus will keep for only a few days and even less than that if it is kept outside of the refrigerator.
Those who do not plan on using their asparagus anytime soon will definitely want to think about freezing it. By doing this you will be able to keep your asparagus for as long as you want, though it will be important to store it in an airtight container. When you get your asparagus home you should make sure to cut off both ends and give it a thorough rinse, no matter how you plan on storing it.