How to Debug Sporadic "Error Establishing Database Connection" (DB failure)

I have WordPress installed on a Digital Ocean VPS, and about once a month, my site will go down with the “Error Establishing Database Connection” error. Restarting the server solves the problem.

From my research it looks like this is a failure of the database server. What is a good way to debug this? What should I be looking for? What are some examples of culprits?

This periodic error happens with a couple of my sites, but for one in particular, the front page will load, but that wp-admin page will show the “Error Establishing Database Connection” error.