Just like you'd store flowers in a vase, asparagus stays fresher when it's standing up in a container of shallow water. Trim an inch off of the bottom of the stalks first, then place the bundled asparagus in something like a mason jar or plastic quart container. Add 1-2 inches of water, cover with a clear plastic bag or plastic wrap, then refrigerate for up to a week. If the water gets cloudy during that time, you can change it out as needed.