How to use "It ain't over till the fat lady sings"?

I know the meaning of this phrase: One should not assume the outcome of some activity (e.g. a sports game) until it has actually finished.

I'm curious as to whether it would more likely be used when

• the speaker refers to an assumed favorable outcome that might turn unfavorable:

"The Giants are going to win this game, no question!"

"It ain't over till the fat lady sings."

• the speaker refers to an assumed unfavorable outcome that might turn favorable

"Might as well go home, no chance the Giants are going to win this one."

"It ain't over till the fat lady sings."

Or does the phrase apply to both situations equally?

I'd say that both of those usages are fine and that you could generalize it to "an unexpected outcome might occur instead of an expected one."

I've also heard it used (on TV) to simply mean that the end hasn't arrived yet and you have to wait for it (before you can, for example, go home).

It refers to either. The meaning of the phrase is that the outcome is still undecided, and whoever is ahead at the moment may still lose, irrespective of which team you are rooting for.