They want to believe!

 The Wright brothers
Some believe the Wright brothers did not in fact build the world's first successful aeroplane. German-born Gustav Whitehead should get the credit, they say, as evidence suggests he took to the sky two years earlier. They add that US institutions will not accept Whitehead's role in the birth of aviation because of their indebtedness to the Wrights' legacy – a 1948 contract between the estate of Orville Wright and the Smithsonian museum means it is legally obliged to call the Wright brothers the first to fly.

 Amelia Earhart
The pioneering aviator Amelia Earhart vanished over the Pacific Ocean in 1937 while attempting to circumnavigate the globe. Various reasons have been given for her disappearance. Some claim she was a spy, and that she was shot down and captured by Japanese forces; some believe she faked her own death; and a few even claim she was abducted by aliens. Earlier this year researchers claimed they had discovered remnants of her aircraft using sonar readings.

 The Bermuda Triangle
The roughly triangular area bounded by Miami, Bermuda and Puerto Rico is where dozens of aircraft and ships are said to have vanished in unusual circumstances, with the disappearances attributed to paranormal or extraterrestrial activity. Notable incidents include the disappearance of Flight 19, a US Navy bomber, on December 5, 1945, as well as the aircraft sent to search for it; that of a Douglas DC-3 aircraft with 32 people on board in 1948; and a mid-air collision between two US Air Force planes in 1963.
 TWA Flight 800
Trans World Airlines (TWA) Flight 800 crashed off the coast of East Moriches, Long Island, on 17 July 1996 en route from JFK International Airport to Paris, France, killing all 230 people on board. After a four-year investigation the official verdict was that the crash was caused by a fuel tank explosion due to a spark from a damaged wire. But not everyone believes that, and the producers of a documentary that aired in 2013 argued that the plane was shot down. FBI investigators said they found explosive residue on the plane, which may have indicated a terrorist attack. The documentary argued the truth was covered up either by the US Government or by the air industry, which might otherwise have faced billions of dollars in financial losses.

 Aerolinee Itavia
Aerolinee Itavia Flight 870 was a domestic Italian commercial flight en route from Bologna to Palermo when it crashed into the Tyrrhenian Sea between Ponza and Ustica, killing all on board in 1980. The disaster led to numerous investigations, legal actions and accusations, and continues to be a source of speculation, including claims of conspiracy by the Italian government and others. Former Italian President Francesco Cossiga officially attributed the cause of the crash to a missile fired from a French Navy aircraft. The conspiracy theories that sprang up after the official verdict included a terrorist attack, a missile strike during a Nato training exercise and that the plane was shot down by a missile in a strike against Libyan forces. In the months and years after the crash seven Italian air officials who were involved in the investigation died in suspicious circumstances including car crashes, murders and suicides.

     STARDUST
Star Dust was a civilian version of the Lancaster bomber and was registered to the British South American Airways. In 1947, it was traveling from Buenos Aires to Santiago, Chile but disappeared over the Andes minutes before its scheduled landing. In his last moments, the aircraft's radio operator transmitted one final, puzzling Morse code message: “STENDEC.” The meaning of STENDEC is still unknown, and has been interpreted in many ways. One string of theories explains STENDEC to be one of a number of acronyms like "Starting En-Route Descent" or "Severe Turbulence Encountered Now Descending Emergency Crash-Landing", but none of these acronyms can be reliably documented for the era.
The vanishing act happened at a time of considerable political turmoil in South America. Deteriorating Anglo-Argentine relations held intriguing implications for the contents of the diplomatic bag carried by the King's Messenger; sabotage might have been a convenient way to ensure that it never arrived at its destination.
Fifty years later, hikers discovered the plane's wreckage in a melting glacier. Star Dust went off course due to then unknown phenomenon jet stream, and in low visibility descended into the cloud-covered mountain range.

Source: telegraph.co.uk, uk.news.yahoo.com, damninteresting.com, listverse.com