How do orphans receive last names.

I know today it's unlikely for an orphan's original parentage to be completely unknown, but if a baby is found and no relatives can be found, how is a surname given?

I've heard that one of two things usually happens (though this assumes the child is an infant, older children would know their names):

1. Surname is given based on where the child was found or something along those lines.
2. In many countries, people can legally change their surname, so they themselves can pick upon turning 18.

I imagine whoever adopts the kid would give them their surname.

Not all get adopted sadly, and before adoption.