It is important that you monitor the MySQL error log. There are a few different options available for defining the details of the log. If not specified the default is [datadir]/[hostname].err. This is not an ideal location for 2 reasons.
First, a correctly configured MySQL datadir will have permissions only for the mysql user, and this is generally restrictive access to the user only and no group or world permissions. Other users/groups should have limited access to the mysql error log.
Second, the datadir is for data, not logs, especially logs that can potentially fill file systems. I am referring here to more then just the error log.
I would recommend you create a separate directory for MySQL logs such as the error, slow and general logs. An example I implement for single installation environments using Linux mysql packages is: