Moderators on S[OFU] have a number of privileges above and beyond those of normal users. These are necessary in order for them to perform their duties as moderators: keeping the peace, identifying and mitigating abuse, etc.
However, most moderators also participate on these sites in a fashion similar to ordinary users: asking and answering questions, voting on posts, etc. And it occurs to me that while so engaged they might occasionally find their moderator tools useful for purposes not directly connected to their jobs as moderators of the site... For instance:
Moderators can view the email addresses of users who have provided one in their profile. Upon answering an unclear question and encountering disagreement from other users who interpreted the question differently, they might find it expedient to email the user directly rather than or in addition to leaving a comment on the question itself.
Moderators can view full names (if specified) and IP addresses, potentially allowing them to track down the physical location of a user. If a question has been edited by its original author such that the answer originally provided by a moderator no longer applies, the moderator could dress himself in a feathered suit and dance in front of the OP's house, creating a spectacle for the neighbors.
Moderators can see who leaves votes on posts. A moderator whose question is down-voted without comment could find out which user had done so, and ask them privately for an explanation.[edit by Jeff Atwood: not true -- striking this out because it borders on dangerous misinformation. See my answer, below.]
Moderators can leap tall buildings in a single bound. A moderator who found himself unable to answer a question due to the interference of a tall building could leap over it.
Note that I'm not claiming any of this actually occurs on S[OFU]... Merely that it could. And here's why I think it should:
Q&A on SO is intended to be valuable beyond a single Asker-Answerer interaction: a good question and corresponding answer can be valuable long after the original authors have moved on. While it might seem unfair that moderators would be able to obtain improvements to questions and/or promote their own answers in situations where other users are not, the end result justifies the means... If normal users could be trusted with the means, they would be allowed, nay, expected to achieve these ends as well!
In order to be fair to other users, a moderator must first be true to himself. If a moderator sees another user a user down-voting his answers without explanation, he might be tempted to retaliate, at which point guilt might prevent him from fairly resolving the disputes of other users. Rather he should sate his curiosity and move on, perhaps even encouraging another user to do a better job of participating in the site.
Leaping tall buildings is cool, regardless of the motivation.