At least in TA, and I think in earlier versions, ship designs are remembered and available for your next game, but with a significant catch. The catch is that up until you save, quit and later reload the game, the designs are remembered and become available when you've researched all the techs needed to build them. After you quit, restart, and reload your saved game, however, those saved designed are no longer available to you.
As DivineWrath pointed out earlier, you can save a stripped ship design as a "template" (there's a "save design as template" checkbox), and then in future games view and "use" that template in the ship designer (click on the template tab). Ships saved as templates are not affected by the save/reload problem I mentioned above.