It’s better to have too much icing than not enough, so I almost always have royal icing left over after finishing a project. Even when I’m going to be using the same icing the next day to finish up a set of cookies, I never store the icing in the bags overnight. I always empty the bags into an air tight container and put the icing in the refrigerator. When I was in culinary school, we would put a damp paper towel over the icing inside the container so that it wouldn’t form a crust. I don’t do this now because the moisture from the paper towel changes the consistency of the icing. As long as you clean the inside edges of the container before closing it up tightly, it will be fine.
The icing will separate after sitting for several hours, so you’ll need to give it a good stir before using it again.
If you won’t be using the icing within 10 days, you can store it in the freezer. I’d recommend not storing it longer than a month, just to be on the safe side. Freezer burn does not taste good.