Everyone makes mistakes. Non-commissioned officers and officers have come to expect it from low-ranking privates, but even with over ten years in the service, you're not exempt from the occasional goof. These accidents range from a mistake in uniform, leaving a CAC in the computer, and anything that falls under the category of "humans making human mistakes."
Private Joe Schmoe has every right and responsibility to make on-the-spot corrections, even to the Chief of Staff of the United States Army. Leaders worth their weight in salt will take the correction and actually respect the subordinate for making it, but only if the mistake is addressed with tact. If you're a Private and you interrupt the Command Sergeant Major because you saw him take two steps while he's on the cell phone — I mean, yeah, you're not entirely in the wrong, but no one will ever see it that way, especially the Command Sergeant Major.
This list outlines the ways you can tactfully correct your superior, starting with the most subtle methods intended for common mistakes and working its way up to grievous errors, with examples for each. Think of these as an escalation of force appropriate to the situation. With respect to the rank of the person being corrected, you should obviously not reach for the sledgehammer tactic to deal with a thumbtack problem.
5. Quietly point out the mistake
Example: Your superior has their patches on the wrong side.
As odd as it sounds to older Army vets and troops from nearly every other branch, a common mistake soldiers make when dressing in the morning is to put the Velcro "U.S. Army" and name patches on the wrong side. This usually happens when someone is in a rush in the morning and it simply slips their mind.
If your superior's made this goof, get their attention and point to your own patches. They should (probably) get the hint.
4. Point out the regulation
Example: Your superior instructs a class incorrectly.
This is best used when they're so confident, but they're so wrong. Don't be a dick about it — you don't need to do the, "well, actually, Sergeant, according to... you're wrong!"
Only attempt this if you're absolutely positive that you're right. If you're only 99.9% sure, start what you're about to say with, "Pardon me, sir, I believe it's..." That way, even if you're wrong, it gives them the opportunity to learn the proper way and you won't be completely oblierated.
3. Pull them aside
Example: Your superior is slacking off.
If you need your supervisor to do something, the most effective way to get them off their lazy ass is to convince them that it's their idea. Use phrases like, "Can you teach me how to..."
Whatever you do, never come at them like you outrank them. You still need to show respect to their rank, even if they aren't acting like it.
2. Inform their peer
Example: Your superior might be drunk on duty.
For better or worse, the military handles issues at the lowest level possible. It's terrible when that policy covers up something that should probably be addressed, but the consequences are the same and it keeps a clean paper trail.
If there's an egregious situation at play that your superior won't or can't address, inform their peer. Pass the concern up the chain of command to someone more appropriate to handle the situation.
1. Inform their supervisor (or MP)
Example: Your superior does something that brings discredit upon the armed forces.
These are your heinous acts and criminal offenses. If they are your superior and you are aware that they did something horribly wrong, do not cover for them. The military justice system doesn't care for the "snitches get stitches" mentality.
If you're aware of criminal activity and you don't speak up, you're guilty as well. All it takes is an open-door counseling to at least one superior to keep you from getting caught up in their crime.