The USS Dixon (AS-37), a submarine tender, was sunk by American carrier aircraft, and armed P-3 patrol aircraft. Navy explosives experts were able to sink the USS Merrill (DD-976) and Leftwich (DD-984) off the Hawaiian islands as part of a training exercise. And one they will probably never forget, as it is rare for EOD (Explosives Ordnance Disposal) folks to practice on ships, and sink them as well. The remaining ships were sunk by gunfire. Missiles and torpedoes, because they are so expensive, are rarely used for these sinkings. The ships are usually sunk in very deep (as in 10,000 feet or so) water.
Not all retired warships are scrapped. Many are instead used for target practice, or to test new weapons and their destructive power. In 2003, the U.S. Navy disposed of 15 decommissioned warship in this fashion. Some of them were destroyed by allied navies that had received used American warships, and were now done with them. These ships are often not given to the foreign navy, but transferred, and, technically, remain the property of the United States. But rather than return a frigate and destroyer, the Greek navy was allowed to use the ex-USS Trippe (FF-1075) and ex-USS Richard E. Byrd (DDG-44) for target practice off the island of Crete (the resting place of thousands of warships over the last four thousand years). The Taiwanese navy also sank the ex-USS English (DD-696).