Natural camera sway

I want to create natural camera sway without animation on my camera to prevent the camera from just being still all the time. I want the sway to be subtle but noticeable and possibly random if that looks better.

How could I do this with a script?

There is a script to do this in the Standard Assets. Have a look at the camera package, and within that, look at the handheld camera. It has some code which attempts to emulate natural hand-held camera swaying.