Yes, we have MySQL (mysql.eng.utah.edu) available for student use.
To request a database, send an e-mail to email@example.com and let us know you need one. We will create it and likely place a file in your home directory with the database name and your login credentials.
The databases are only accessible directly from the Linux machines in CADE (‘mysql -u username -h mysql -p’), although if there is a need, we can open them up to the Windows machines. If you place website files in your .public_html directory, you can make calls to the db, and thus access it from anywhere via www.eng.utah.edu/~username.