Flat badges are meant for wallets, cases or displays, such as shadow boxes or plaques. In fact, sometimes a wallet badge is called a "Flat Badge"! Usually the attachment for a flat badge will be either Wallet Clip or "Glue in Wallet'.
Curved badges are meant for uniforms. Curved badges usually have a pin & safety catch attachment. Curved badges can make wallets extra bulky, so they are not recommended. Treat yourself to a dedicated "flat badge" instead!