What's the best way to store asparagus?
In a cup with water. You place them straight up in a small cup with a little water. Just like you would if you were storing flowers. 

You'll want to store this in your chill chest on the top most rack; to avoid any cross contamination from other possible food products (chicken, etc).

You may want to leave the rubber band on to allow it to stay tidy. If you must you can cover your asparagus cup with a zip lock bag on top. 

With this method we usually can store asparagus for 5-7 days.

If you must keep them longer it maybe better to just freeze them in an air tight bag. after they're dried out.