Hats There are hats available to buy in the southern empty house. When you get your first achievment a mouse will move in and you will be able to buy your first hat. Hats are purely cosmetics.
Rings there are a few rings in the game. Magnet rings increase your pick-up radius. Golden rings increase light-radius around you. There are also rings that improve attack speed and crit chance which are buyable later in the game in the adventurer guild.