A savvy admin will pick battles carefully. I've seen too many cases of "you're right, but you're wrong to bring it up" over the years. So as a public service to any newbie administrators out there, I'll offer a few examples of battles you're really better off skipping, if you have the option.
1. The photocopier. You will never win the battle of the photocopier. Yes, Professor X will beat the poor thing into submission, running off copies like he has a grudge against trees. And yes, other professors who teach the exact same class will somehow manage not to. Yes, toner costs money, as does paper, and the environmental damage from excessive copying is real. But you will not win this one. I've seen too many deans or chairs come to grief trying to hold the line on copier costs. Prof. X will immediately leap to academic freedom, claiming that your inquiry into his thousands of copies is a thinly-veiled attack on his choice of instructional materials. Other faculty who normally think of Prof. X as a bit of a pain will rally to his side, out of fear that you'll start charging them by the page. No good can come of this. (Exception: if you have a department secretary who is longstanding, respected, and ferocious, s/he might be able to pull it off. But you won't.) The only concession I've seen successfully wrested on photocopying has been a grudging willingness to do double-sided, to save paper. If you get that, call it a win and move on.
2. "So-and-so doesn't attend the meetings of the committee he signed up for." This falls victim to the "what about everybody else" problem. Unless you're willing and able to do surveillance on every single committee, and to cross-reference every absence, you're typically best off leaving this one alone. Yes, it's unfair to the 'good soldiers,' but any system built on honor and discretion will have some holes in it. Given the distastefulness of the alternative, you're typically best advised to let this one slide. If the faculty want to own governance, let them own non-participation, too.
3. "It's unfair that Prof. X is such a harder grader than Prof. Y!" This is another of those "yes, but" situations. Yes, it's unfair (if/when it's true), but you don't want to go there. This is why colleges have Procedures. Refer the student to the Grade Appeal Process, and leave it at that. If there's a consistent stream of students about the same professor, talk to the department chair about grade norming workshops. But don't expect much. This one combines the academic freedom issue with the surveillance issue, making it a no-win twofer. Unless you can show a double standard based on some sort of identifiable discrimination, walk away. Professors are allowed to be strict. Don't take the bait.
4. The decades-old grievance. Let the past pass.
5. "Everybody knows." You'd be surprised how difficult it is to actually prove what "everybody knows" to be true. Widespread belief is not proof. Be prepared to be accused of condoning whatever it is that everybody knows. ("The Administration knows about it, but doesn't do anything!") In the absence of actual proof that you could defend in court, hearsay is just hearsay. In my observation, don't be surprised if the very people urging you to action scatter abruptly when asked to sign their names to a complaint. It's called cowardice. Walk away.
Wise and worldly readers -- what would you add?