You know how when you name a shulker box its name is visible above the inventory when you open it?

If a chest has a sign on it, the text (as much as will fit, perhaps) should be visible above the inventory, too.

So if you have a chest with a sign on it that says "cobblestone", when you open it you can see the text of the sign as well as the contents of the chest.

If there is more than one sign on the chest it could default to the sign on the left-front, which on 1x1 chests would be just the sign on the front.