Flat badges are meant for wallets, cases or displays, such as shadow boxes or plaques.
Curved badges are meant for uniforms.
A slightly curved badge is somewhere between these two. Smith & Warren tries very hard to meet this specification, but it can be difficult, so be aware that your badge may be more on the curved side in some cases.
Curved badges can make wallets extra bulky, so make sure you know where your badge will be displayed before you select the Shape for custom badge orders.