Brussels sprouts will keep for quite a while once they've been picked - at least several weeks and into the New Year. This was good news to us since we're big fans of sprouts and mourn the brevity of their appearance at the farmer's market right before it closes.
Go ahead and take the sprouts off the stalk, but leave all their outer leaves intact. Store them in the fridge in a bowl or unlidded storage container. The outer leaves will shrivel and wilt in the open air, but the inner (and yummy!) part of the sprout will remain protected. Remove those outer leaves before cooking.
Have you ever stored brussels sprouts through the winter?
Related: How to Store Apples for the Winter