In most European countries the Ultras are in charge of support. Some group comes up with a new song and the other people will start to sing it too eventually. Sometimes they even hand out small notes with the text for a new song. But also some songs go back decades and it's often forgotten how it started. Melodies are usually from famous songs e.g. Yellow Submarine, so it's pretty easy to memorize.

Since they don't have Ultras in England I can't really tell how it works there. I guess a bunch of people come up with something and if it's catchy it will spread quickly.