If you don't see a design you own available to build in the Shipyard, you need to set it as your default design in the Ship Design Center building.
Additionally, later ship types (e.g. Destroyer, Light Cruiser, Battle Cruiser, etc.) are only unlocked by upgrading the Ship Design Center. If you don't have a ship type unlocked, you will not be able to build any variants of it until you upgrade your Ship Design Center to the appropriate level.