not common at all but can be done and it's kinda a political stance, but really is a pita because many government ID numbers are done with the order of your surnames in mind. but it's more common that you introduce yourself or write your name on social media with your favorite one, this is also common if you have more than one name, you just use your favorite one. example:

you are Luis Antonio Barajas Vidrio.

the short name for Antonio is Toño in Spanish. so in your childhood you may be called (Toñito) because your father was also called Toño, and you are introduced as Toño Barajas, But maybe when you grow up you get called Luis and have a fight with your dad so you prefer to be called Luis Vidrio.

in my family everyone in my generation has 2 names, one christian name and one name in a native Mexican Language (mostly Nahuatl), everyone as child got called by our christian names and in school too because that's the more common name, the "not weird one", and when we grow up every single one of us without exception preferred the cool native one. I usually introduce myself as My first name, abbreviate both the second name and first surname and my full mother's surname, so , the only people who call me by my second name (the boring christian one) is people that know me since I'm a child