Wordpress does not support anything other than MySQL, which is somewhat unique as far as CMSes go. There seem to be a few Drupal addons that don't mix with Postgres, but other than that it should be fine. Same with all others.

Generally, if you can, you should consider supporting Postgres in your framework. It's a much saner and robuster database from an Ops point of view (replication doesn't fail as often) and more flexible from a Dev view (performance is generally more predictable, much wider feature set with document storage / PostGIS / etc., …).

I like Postgresql, but it's far from superior to MySQL from an ops monitoring point of view. MySQL exposes a lot more information and run-time stats than Postgresql does. Simple things in MySQL are also impossible in Postgresql. How do you simply and effectively guarantee that your Postgresql replicas are not lagged and are indeed connected to the master and successfully writing down new logs in a streaming hot standby replica? You can, but it requires that all of your replication accounts also have superuser access on the master. Fun stuff, that.

Which is IMO preferable to MySQL servers randomly losing sync for no reason every few weeks. I don't care how many tuning knobs MySQL has if none of them makes it work as smoothly as Postgres.