How do you clean Boontonware?

It depends what you mean by "clean". If you want to restore the shine/luster and/or remove stains I find using this product works great. Nu-Finish is a silicon based car wax, I recommend the liquid, it comes in an orange bottle and if you have a Wal-Mart near you it's the cheapest source. Don't use the paste wax; my experience is it doesn't work as well. The silicon will clean and seal the plastic and leave a nice sheen. Wash and dry the item first, wear gloves if you're adverse to having chemicals come into contact with your skin. Follow the directions on the bottle and make sure you buff with a clean soft dry rag. After you've "fed/sealed" the melamine, simply wash it with hot soapy water. I've used harsh abrasives to restore the finish of previously "scoured" melamine with several methods and products with mixed results (it comes clean but the finish is badly scratched from things like a salt/lemon mixture or soft scrub). A baking soda paste can be useful if you use a cotton rag and carefully avoid creating swirls. Keep in mind that melamine is a hard plastic material that cannot be microwaved. When I use my Bootonware, I only serve dry goods in them; popcorn, chips, cookies etc, nothing that could leave a stain or would have to be cleaned aggressively.