How powerful is POSIX regex
In [1], the authors formally define the notion of an "extended regex" with the intent of capturing the back-reference capability of POSIX/perl/emacs/etc style regexes. Exactly how closely their definition matches the exact POSIX specification is an exercise left to the reader. 

Under their definition, extended regexes are a proper subset of Type 1 (context-sensitive) languages and incomparable with Type 2 (context-free) languages. They show that extended regexes are a subset of Type 1 languages by showing how to construct an equivalent LBA for any extended regex. They prove that they're incomparable with Type 2 languages by showing that $\{a^nba^nba^n|n \geq 1\}$ (which is not context-free) can be recognized by an extended regex while $\{a^n b^n | n \geq 0\}$ (which is context-free) cannot.


Câmpeanu, Cezar, Kai Salomaa, and Sheng Yu. "A formal study of practical regular expressions." International Journal of Foundations of Computer Science 14.06 (2003): 1007-1018.