In my experience "J Smith" is used more often than "John S". For example, when quoting someone it's common to just use initials for all their names, except the surname. e.g. "C. S. Lewis".
I haven't come across "John S" very often. I guess someone might use it for confidentiality reasons... e.g. reception staff at my work do that on their name badges.
Generally calling someone by their first name is more friendly/informal, whereas calling them by their surname is more formal. What you use in your own signature is personal preference but it's worth knowing this general convention so that you come across how you are intending (e.g. in a job application I'd use both my first name & last name, whereas in a email to a colleague, just my first name).