May I ask what you do use exactly? MySQL has always been my goto for simple db needs. What do you normally use and what's your "general" cases where you wouldn't use it?
For "simple [rdms] needs" I would recommend SQLite. For anything more, PostgreSQL. I find MySQL too buggy and it diverts from the SQL standard too often (or doesn't implement enough of it) for my tastes. A properly configured postgre install (granted, not exactly trivial) will perform at least as good if not better than MySQL and its advanced functionality is extremely mature and robust, unlike MySQL's which has largely been tacked on in the current major version (views, templates, triggers, etc.)