PostgreSQL does natively support POSIX-style regular expressions, which can do advanced stuff like captures, positive and negative lookahead (alas, no lookbehind), etc.

See also here: docs Pg pattern-matching (be sure to read past the simple LIKE expressions! )

PostgreSQL does support regular expression indexing, which is powerful, but it has some limitations too. See the 'text_pattern_ops' operator class (here). (generally: Pg manual, chapter 11)

Pg also has full text indexing, which can index on words and their derivations (but no regexen).