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With all leaf lettuces, I wash when I get them home. A key step is to dry thoroughly, to the point of feeling like you are over-doing it. Then pack lightly in a plastic bag with strips of paper towel to absorb and protect the leaves. Smush out all of the air and close the bag. I store in the vegetable bin in the fridge. The lettuce will last about a week, though we usually eat it all up before then.
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I often store it much as I do parsley. Separate the bunch and wash it well to get rid of the inevitable grit. Spin it as dry as possible in a spinner, and blot any large droplets with paper towels. Then take a long strip of paper toweling and spread the arugula (or parsley) in a single layer on the toweling. Now, roll it up, but not to tightly, and put the roll into a zip lick storage bag with the top open. Store this in your crisper. The arugula/parsley should be good for at least a week (or until you use it up.
I do the same washing and thorough drying, but instead of a plastic bag, I store it in a large tupperware container lined with a paper towel on the top and the bottom. I do this for all salad greens and they last forever this way!