when can we use "flared/flaring nostril?"

Whenever you want to use it. Without context it'd be hard to say.

the moment he heard his name among the nominees, he started flaring his nostril

There's nothing wrong with saying that.
However, I have a slight preference (shorter) for saying,

The slight technicality...is that a flared nostril is often more of a unconscious reaction to something...hence perhaps it could have a passive tone. That is my opinion.

Also we might imagine "flared" referring to a more nearly permanent condition, a "past tense" so to speak

This is the one I'd opt for - only I'd say his 'nostrils' flared - jolly hard to flare just one nostril!

I can do it. Watch...

I just checked. I can raise one eyebrow and lower the other, but one nostril is beyond my.

That's why I like 'ya, Rog! You gave it the old college try.