All of these events were found on Wikipedia by searching for the numerical year, and then scrolling to the month of May on the summary page that resulted.
May 1961 - Alan Shepard becomes the first American in space aboard Mercury-Redstone 3. Had the flight taken place on its original scheduled date of March 6, 1961, Shepard would have been the first man in space.
May 1962 - Twelve East Germans escape via a tunnel under the Berlin Wall.
May 1963 - Thousands of African Americans, many of them children, are arrested while protesting segregation in Birmingham, Alabama. Public Safety Commissioner Eugene “Bull” Connor later unleashes fire hoses and police dogs on the demonstrators.
May 1964 - John George Kemeny and Thomas Eugene Kurtz run the first program written in BASIC (Beginners’ All-purpose Symbolic Instruction Code), an easy to learn high level programming language which they have created. BASIC is eventually included on many computers and even some games consoles.
May 1965 - Forty men burn their draft cards at the University of California, Berkeley, and a coffin is marched to the Berkeley Draft Board.
May 1966 - The Communist Party of China issues the ‘May 16 Notice’, marking the beginning of the Cultural Revolution.
May 1967 - Tennessee Governor Ellington repeals the “Monkey Law” (officially the Butler Act; see the Scopes Trial).
May 1968 - The Beatles announce the creation of Apple Records in a New York press conference.
May 1969 - An American teenager known as ‘Robert R.’ dies in St. Louis, Missouri, of a baffling medical condition. In 1984 it will be identified as the first confirmed case of HIV/AIDS in North America.
May 1970 - Kent State shootings: Four students at Kent State University in Ohio are killed and 9 wounded by Ohio State National Guardsmen, at a protest against the incursion into Cambodia.
May 1971 - The birth of Bangladesh is declared by the government in exile, in territory formerly part of Pakistan.
May 1972 - The Watergate first break-in, the “Ameritas dinner”, fails.
May 1973 - A 71-day standoff between federal authorities and American Indian Movement activists who were occupying the Pine Ridge Reservation at Wounded Knee, South Dakota, ends with the surrender of the militants.
May 1974 - Dublin and Monaghan bombings: The Protestant terrorist group, the Ulster Volunteer Force (UVF), explode numerous bombs in Dublin and Monaghan, in the Republic Of Ireland. The attacks kill 33 civilians and wound almost 300, which is the highest number of casualties in any single day during “The Troubles”.
May 1975 - Junko Tabei becomes the first woman to reach the summit of Mount Everest.
May 1976 - Ulrike Meinhof of the Red Army Faction is found hanging in an apparent suicide, in her Stuttgart-Stammheim prison cell.
May 1977 - Star Wars opens in cinemas and subsequently becomes the then-highest grossing film of all time.
May 1978 - A bomb explodes in the security section of Northwestern University, wounding a security guard (the first Unabomber attack).
May 1979 - Counting in the previous day’s British general election shows that the Conservatives have won and Margaret Thatcher becomes the country’s first female prime minister.
May 1980 - Mount St. Helens erupts in Washington, killing 57 and causing US$3 billion in damage.
May 1981 - A jury of architects and sculptors unanimously selects Maya Lin’s design for the Vietnam Veterans Memorial from 1,421 other entries.
May 1982 - Cal Ripken, Jr. plays the first of what eventually becomes his record-breaking streak of 2,632 consecutive Major League Baseball games.
May 1983 - Stern Magazine publishes the “Hitler Diaries” (which are later found to be forgeries).
May 1984 - The Soviet Union announces that it will boycott the 1984 Summer Olympics in Los Angeles, California.
May 1985 - .S. President Ronald Reagan joins German Chancellor Helmut Kohl for a controversial funeral service at a cemetery in Bitburg, Germany, which includes the graves of 59 elite S.S. troops from World War II.
May 1986 - Hands Across America: At least 5,000,000 people form a human chain from New York City to Long Beach, California, to raise money to fight hunger and homelessness.
May 1987 - Nineteen year-old West German pilot Mathias Rust evades Soviet air defenses and lands a private plane on Red Square in Moscow. He is immediately detained (released on August 3, 1988).
May 1988 - Soviet war in Afghanistan: After more than 8 years of fighting, the Red Army begins withdrawing from Afghanistan.
May 1989 - The first crack in the Iron Curtain: Hungary dismantles 150 miles (240 km) of barbed wire fencing along the border with Austria.
May 1990 - The World Health Organization removes homosexuality from its list of diseases.
May 1991 - Queen Elizabeth II becomes the first British monarch to address the United States Congress.
May 1992 - STS-49: Space Shuttle Endeavour lands safely after a successful maiden voyage.
May 1993 - Juan Carlos Wasmosy becomes the first democratically elected President of Paraguay in nearly 40 years.
May 1994 - Illinois executes serial killer John Wayne Gacy by lethal injection for the murder of 33 young men and boys.
May 1995 - U.S. President Bill Clinton indefinitely closes part of the street in front of the White House, Pennsylvania Avenue, to vehicular traffic in response to the Oklahoma City bombing.
May 1996 - Everest disaster: A sudden storm engulfs Mount Everest with several climbing teams high on the mountain, leaving 8 dead. By the end of the month, at least 4 other climbers die in the worst season of fatalities on the mountain to date.
May 1997 - U.S. President Bill Clinton issues a formal apology to the surviving victims of the Tuskegee Study of Untreated Syphilis in the Negro Male and their families.
May 1998 - The first euro coins are minted in Pessac, France. Because the final specifications for the coins were not finished in 1998, they will have to be melted and minted again in 1999.
May 1999 - Microsoft releases Windows 98 (Second Edition) (from 1998).
May 2000 - The billionth living person in India is born.
May 2001 - Space tourist Dennis Tito returns to Earth aboard Soyuz TM-31. (Soyuz TM-32 is left docked at the International Space Station as a new lifeboat.)
May 2002 - Former U.S. President Jimmy Carter arrives in Cuba for a 5-day visit with Fidel Castro, becoming the first U.S. President, in or out of office, to visit the island since Castro’s 1959 revolution.
May 2003 - U. S. president George W. Bush lands on the aircraft carrier USS Abraham Lincoln, where he gives a speech announcing the end of major combat in the 2003 Invasion of Iraq. A banner behind him declares “Mission Accomplished”.
May 2004 - Massachusetts legalizes same-sex marriage, in compliance with a ruling from the state’s Supreme Judicial Court (Goodridge v. Department of Public Health).
May 2005 - Kuwaiti women are granted the right to vote.
May 2006 - Human Genome Project publishes the last chromosome sequence, in Nature.
May 2007 - The Russian Orthodox Church Abroad and the Moscow Patriarchate re-unite after 80 years of schism.
May 2008 - Over 69,000 are killed in central south-west China by the Wenchuan quake, an earthquake measuring 7.9 Moment magnitude scale. The epicenter is 90 kilometers (55 miles) west-northwest of the provincial capital Chengdu, Sichuan province.
May 2009 - North Korea announces that it has conducted a second successful nuclear test in the province of North Hamgyong. The United Nations Security Council condemns the reported test.
May 2010 - Scientists conducting the Neanderthal genome project announce that they have sequenced enough of the Neanderthal genome to suggest that Neanderthals and humans may have interbred.
May 2011 - Osama bin Laden was shot and killed inside a secured private residential compound in Abbottabad, Pakistan, by U.S. Navy SEALs in a covert operation orchestrated and authorized by U.S. President Barack Obama.