They're used when you hit your "fire" button for the first time. The little red box in the top left will tell you if they're ready or not (Filled = yes, Empty = no). Some will just fire once and recharge (stuff like Soul of Zinglon) after a while, others need to mash to work (Repulsor, for instance).

They won't work unless you let go and start firing again, which is probably why you couldn't figure it out.

You can also use Enter in Super Arcade Mode (Rear Weapon Switch) key for the code ships to change which Special they have.