I talked to a friend who said it was considered a "bad habit", or "bad style" for a dancer to start humming, sing along (if not performing in a musical of course) or mouth along to the music while dancing.
If we're talking about social dancing, then I
strongly disagree with your friend!
In my opinion, the whole
point of social dancing is to enjoy the music. So, I actually take it as a high compliment when my partner is singing along with the music, because that means my lead is non-intrusive and natural enough that it doesn't interrupt her ability to focus on enjoying the music in whatever way makes her most comfortable -- including singing. In other words, she doesn't have to waste time concentrating on how to follow my moves, but instead she can simply concentrate on the music while the moves just magically "happen."

I also do my best to reduce any stress or self-consciousness in my partners, so singing is another indication that she has let down some of the social barriers that might otherwise exist between two people.
So, ladies, sing away when you're dancing with Big10!
it's true, the feeling of being watched is very offputting and makes it hard for me to lose myself. i feel this more when my students are there, cuz
i feel like i shouldnt make any technical errors 'practice what you preach' and all that...
I teach a couple of classes, too, but I always give new people the "do as I say, not necessarily as I do" speech before our studio socials.

It helps that our studio markets itself as teaching "street" Salsa, thus reducing some of the expectations of constant formality. Plus, from a personal standpoint, I try not to dance "like a teacher" when I'm out socially. Maybe that's not the best strategy from a marketing perspective by making sure that onlookers realize they're watching a teacher from whom they can take lessons -- but it does give me added credibility when I happen to talk to somebody and show how they can grow (like me) from a "gringo" into dancer who looks somewhat "authentic" or like he/she has a background social dancing in Latin countries.