In Superman's case, I don't even see the need for a secret identity, he's Superman!!
He should just say; "Hi the name's Clark, just provide me all my basic earthly needs and I'll save your asses every five minutes. And I don't work weekends."
I don't know, it would be hard to have any kind of meaningful personal life if everyone knew who he was.
I'm willing to bet just about any celebrity today who is relentlessly hounded by the paparazzi would love to have a secret identity which would allow them to function normally in society.