I have a plastic tub with hundreds of bottle caps and corks in it. I also have a shadow box with some of my favorites inside.
But if you want to show pride in your home state or your favorite brewing state, then might I suggest Beer Cap Maps. Lovely wood in the shape of a state or the whole US that you can plug your bottlecaps into.
I see them as perfect for your part of the home that is dedicated to beer décor. Much prettier and less gaudy then a neon sign.