Imagine a situation in which the whole place is on fire, a bomb is about to explode, everyone is running for their lives and someone is checking his looks on the mirror... pretty inappropriate for the situation, don't you think? I'm looking for an idiom or colloquial/slang/informal expression to describe such an indifferent stance.
In software engineering the term "bikeshedding" is used to describe this situation.
For more information read the Wikipedia entry for Parkinson's Law of Triviality.
I've heard straining at a gnat, but swallowing a camel.
I always liked the quote from the musical "The Music Man":
There's a burglar in the bedroom while you're fiddlin' in the parlor!
Seems to fit fairly well.
If the person's nonchalance in the face of danger is due to informed confidence that the threat will be taken care of by others, I'd say they are "cool as a cucumber" or "steely-nerved".
If, on the other hand, it is due to irrational denial or dismissal of the threat, I'd say they are "bonkers", "loony", or some other synonym for "out of their gourd".
If, on the third hand, their lack of response is due to distraction or inattention that is keeping them from perceiving the threat, I'd say they are "out to lunch", or "bliv"--a contraction of "oblivious" that has been used as a derisive appellation in my family for decades.