Q:

What's the more appropriate substitution for "give a f**k/d**n/s**t"

I want a more appropriate phrase that has the same form ("give a **") and meaning as the offensive "give a f**k/d**n/s**t". Is there any?

A:

How about not giving a rat's tail?

A:

"I couldn't care less," perhaps?

A:

'give a monkey's' is all that comes to my mind. It is used as a direct euphemism specifically to avoid using the profanities in the examples you offered.
I believe it is shortened from "give a monkey's ass (arse)", which might still be considered offensive by some.

