I would add some of the more common day-to-day operations:

* Reset a user's password.
* Display all print queues which have something queued.
* Remove an item from a print queue.
* Display mail queue.
* Remove item from mail queue.
* Snapshot of current system performance.
* History of recent system performance.
* Show number of users on the system.
* Show who is on the system complete with IP address.
* Display disc free space compared with set warning limits.
* Read system logs (filtering anything normal). Including hardware logs. Including cron log.
* Read root mail. (I archive the mailbox at 12:00 each weekday to make this manageable and ensure that all cron jobs silent unless they fail). Never send the output from a cron job to /dev/null - one day you will want to read it.
* Emergency find of abnormal large file(s) in the event of a disc space crisis. (Can take hours to write because you need to be able to exclude anything normal). This script could save you because you can be quick by being prepared for the inevitable. It is inevitable that you will encounter a rogue program which will attempt to fill a filesystem.

For the day-to-day server checks I get each system to mail me (and other support staff) with a concise summary of all the services (UP/DOWN) backed with full detail in the event of an actual problem. The fundamental check that the hardware map is the same as yesterday is the most important