IIRC, the patriarchal surname (i.e. mother's father's name and father's father's name) is carried on.

I believe it's also tradition that a spouse doesn't change their name but adopts a qualifier, so they keep their paternal surname and append 'de SPOUSE_SURNAME' to signify marriage in general conversation, though they are still formally 'X Y de Z'.

But I'm only an outsider on this, so I could be totally wrong here or missing some semantics or subtleties. I love learning about other cultures but there's so much to them that's hard to grasp without living within the culture for a long time.

In Spain usually the father's name.