Despite being representative of early 1930s aircraft design and teetering on the edge of becoming outdated (in comparison to some alternatives), the Swordfish achieved some spectacular successes during the war. Notable events included sinking one battleship and damaging two others of the Regia Marina (the Italian navy) during the Battle of Taranto, and the famous attack on the German battleship Bismarck, which contributed to her eventual demise. Swordfish sank a greater tonnage of Axis shipping than any other Allied aircraft during the war.[1] The Swordfish remained in front-line service until V-E Day, having outlived some of the aircraft intended to replace it.
The main task of ship-based aircraft was patrolling for Axis submarines and surface-raiders. By March 1941, Walruses were being deployed with ASV radar systems to assist in this.[45] During the Norwegian Campaign and the East African Campaign, Walruses saw limited use in bombing and strafing shore targets.[46] In August 1940, a Walrus operating from HMAS Hobart bombed and machine-gunned the Italian headquarters at Zeila in British Somaliland.[3][47]
By 1943, catapult-launched aircraft on cruisers and battleships were being replaced by radar, which occupied far less space on a warship. Walruses continued to fly from Royal Navy carriers for air-sea rescue and general communications. The low landing speed of the Walrus meant they could make a carrier landing despite having no flaps or tailhook.[48]
Walruses operated against submarines throughout the Second World War, and were also adopted by the RAF Search and Rescue Force to recover personnel from the sea. It was intended for the Walrus to be replaced by the more powerful Supermarine Sea Otter, but this was not implemented. Following the end of the war, the Walrus continued to serve as a military aircraft, and some aircraft operated in a civil capacity in regions such as Australia and the Antarctic. The Walrus was succeeded in its air-sea rescue role by the first generation of helicopters.
The first operations on board an aircraft carrier commenced in November 1940.
Despite the intention to replace the Swordfish, the Albacore served with it and was eventually retired before it, both aircraft having been replaced by a pair of monoplane designs, the Fairey Barracuda and Grumman Avenger.
Apparently the US did, in fact, have a biplane operating on a carrier in WW2, but it was only used on the Hornet for a few months during sea trials:
The Curtiss SBC Helldiver was a two-seat scout bomber and dive bomber built by the Curtiss-Wright Corporation. It was the last combat military biplane procured by the United States Navy. Delivered in 1937, it became obsolete even before World War II and was kept well away from combat with Axis fighters.[1]
They were apparently used on land bases for a few years in the war, though.
Not a complete list – it didn’t list the Walrus – but this is my source for most of the above.