MySQL is fine - I did some plugins that demonstrate using that. However MySQL is not zero-administration like SQLite3 is. In other words, you need the MySQL server to be running somewhere.
For a single process (ie. a MUD world) that just wants to save stuff (like mobs) to a database, probably SQLite3 is the easiest to use.
http://mushclient.com/sql
However if you want to use MySQL (and there is no major objection at all) then see this forum post:
http://www.gammon.com.au/forum/?id=5983
You would need to download and install MySQL, make sure it is running (eg. as a service on Windows, or possibly on a separate PC somewhere) and then interface with it as described in that forum post (this requires the mysql.dll to be installed to provide the functionality).
SQLite3 however is actually coded into recent versions of MUSHclient, and is available without any installation, configuration, or extra DLLs.