How old are you? or What is your age?
Which is more common or used more and also the correct way of asking?
In any spoken English How old are you is far, far more natural than what's your age. So, if you don't want people to notice your stilted English, it is far better to use How old are you?.
It might also be helpful to know that we don't usually use the words years old in the reply. We just state the number. And it's actually ungrammatical to say just years:
There may be special circumstances when someone might say What's your age?, but the normal question is How old are you?. The answer is "(X)", not "(X) years", or even "(X) years old" - unless you are five!