I've cut my nostril twice in some specific place which doesn't stop bleeding for ten minutes, really small cut, but it's very annoying. My facial hair is still insufficient to take part in the Movember though
I might have told this story before but one night I shaved and then went to bed. The next day I had a pain in the inside of my nostril. I realized I left a bit of shaving cream inside my nose overnight and it corroded a small part of my skin.
I dislike shaving my mustache area the most since it hurts the most. Under the chin is probably the least painful. Or actually the sideburn area is.
I know you were referring to cutting yourself though, luckily I don't have an issue with that.
The reason your moustache hurts is because the hair pulls there. Because you can't start from above out easily, and the hair is thick there, it pulls the hair. It also means it's more likely your razor is grabbing the hair from half way up, rather than skin level. Again creating pull.
Directly under your chin and jaw is ok, around your neck and voice box is the worst. You get those two concave areas either side of your voice box, and the convex area over your voice box, all with skin that is thinner and softer than face skin.
Sideburns are probably easiest though, yes.
I guess I was more referring to shaving your moustache should never lead to being cut