This is one of the most famous scenes form "Clockwork Orange." Malcolm McDowell improved this entire scene,  and spontaneously started to sing "Singing in the Rain," from the musical by MGM, which was created in 1952. This song was originally performed by Gene Kelly, and he sings it with great joy, and represents a celebration of life. Malcolm McDowell sings this exact song, but its during the most horrific scene in the movie. This just shows how much has changed from life in the 50's to life in the 70's. In that span of 20 years, many things changed. Women could take birth control and were free to explore sexually. At the same time, many people of the younger generation were experimenting with drugs such as LSD and marijuana. The style of music began to change and people were way more open and creative. There were also a great deal of people who hated the "corporate world" and war, making people rebellious. I think Malcolm McDowell chose to sing this song because it represents a theme of complete joy, so basically he is being sarcastic at the most extreme level possible.