PostgreSQL it is a smooth database.
MySQL is more for those wanting to switch engines, but of course a lot of ISPs bundle it so people who have taught themselves from an ISP account tend to use it.
MySQL use to be the posterboy for opensource databases, but that was because PostgreSQL was less well known back in the day, primarily because it is older in the world of opensource being closer involved to BSD, and didn't get the shiny new Linux look'in that MySQL got.
PostgreSQL is just rock solid, excellent online help both on the web and in psql, and as someone has already mentioned it does adhere to the SQL standards more than any other RDBMS. I prefer PostgreSQL to Oracle and Sybase, far more flexible. As far as functions goes you are offered a number of different systems; PL/Python, PL/Perl, PL/PGSQL, PL/TCL. Personally I prefer plgsql, but if you are au fait with another language you can use that.