She has been arrested by Francis Grimes, possibly on some green card issues, and sent to Mark's class at the Department of Social Services in the episode "Sympathy for the Devil". Mark walks in on her having sex with Randall Skeffington but when Mark tries to tell Great Brain, Dolores attacks Mark telepathically. It turns out, however, that Dolores and Great Brain get off on playing mind tricks on others.
Randall has read about her information on a DOI register form found in Mark's briefcase. It is written that her name is Dolores. Below is a handwritten field title "Surname:" (with the colon) which is empty. The reason for the handwriting has not been revealed. Whether Dolores's surname is really "Surname" or it has been written on the form by someone in the DOI and pending to be filled is not known. Nevertheless, Randall later calls her Dolores Surname and she seems okay with it.