May: Operation Pluto: The plan to construct oil pipelines under the English Channel between England and France is tested in the River Medway.
May 5: Operation Ironclad: United Kingdom forces invade the French colony of Madagascar.
May 7: On Corregidor, the last American and Filipino forces in the Philippines surrender to the Japanese under command of 2Lt. Robert L. Obourn (92nd Coastal Artillery Corps Battalion, G Battery) from Fort Mills.
May 8: The Battle of the Coral Sea (first battle in naval history where 2 enemy fleets fight without seeing each other's fleets) ends in an Allied victory.
May 8/May 9: At night, gunners of the Ceylon Garrison Artillery on Horsburgh Island in the Cocos Islands mutiny. The mutiny is crushed and three executed (the only British Commonwealth soldiers to be executed for mutiny during the Second World War).
May 12: Second Battle of Kharkov: In the eastern Ukraine, the Soviet Army initiates a major offensive. During the battle the Soviets capture the city of Kharkov from the German Army, only to be encircled and destroyed.
May 12: The Japanese minelayer Okinoshima is sunk by the American submarine USS S-42 (SS-153).
May 14: Aaron Copland's Lincoln Portrait is performed for the first time by the Cincinnati Symphony Orchestra.
May 15: In the United States, a bill creating the Women's Auxiliary Army Corps (WAAC) is signed into law.
May 20: The first African-American seamen are taken into the United States Navy.
May 21: Mexico declares war against Nazi Germany after the sinking of the Mexican tanker Faja de Oro by the German U-boat, U-160, off Key West.
May 26: Battle of Bir Hakeim: The Free French and British troops slow the German advance in North Africa.
May 26: Anglo-Soviet Treaty of 1942 to help establish military and political alliance between the USSR and the British Empire is signed in London by foreign Secretary Anthony Eden and by Soviet foreign minister Vyacheslav Molotov.
May 27: Operation Anthropoid: Czech paratroopers attempt to assassinate Reinhard Heydrich in Prague.
May 31-June 1: Attack on Sydney Harbour: Japanese submarines infiltrate Sydney Harbour in an attempt to attack Allied warships. (source: Wikipedia)