Having two last names is still the norm, however its importance has diminished because of the diminished social stigma around bastardy.

Previously, and especially under the more traditional norms of Hispanic culture, having only one last name was a sign that you were illegitimate and thus not permitted to carry your father's last name. Or it was a sign you may have been an orphan, such as the use of Valdés in Cuba to designate orphans (read the contemporary novel Cecilia Valdés for examples of this).

In most contemporary Latin American cultures, however the stigma of only one last name is largely diminished and among those immigrating to the US the point of using two almost disappears entirely, becoming more of an annoyance whenever you have to write/sign your name.

I can speak for Argentina, since I live here.

Two surnames isn't the norm. In fact, very few people (if you take into account total population) use it these days, as most families just put the fathers name on their children. Last august, our Code of Civil Laws (the main source of legislation for civil matters) went through a major reform that (among other things) allowed families to use either the surname of one parent or the other, or even both at the same time, while the old Code compelled families to use the surname of the father first. I heard a legislator saying that this was in hopes of making people put both surnames on their children, in order to reinforce their identity, but I cannot find the source for this.

In short, at least in Argentina, it is rather uncommon to use two (or more) surnames, and speaking as a lawyer and a teacher, I can't think of any of my clients or students that actually has two surnames. Oddly enough, my father (who is a direct descendant of a spanish man, from Andalucía) has THREE surnames (García Álvarez Asencio) while I only have one (García)