A rat's arse is usually something you don't give. So, if someone says "
I don't give a rat's arse!" it's just another way of saying "I couldn't care less!" or "
I don't give a fuck!"
However, it is possible to give a rat's arse, too. Many years ago a meteorological (Met) colleague working at an Antarctic base opened a crate of equipment that had been shipped
to the base many months previous. Inside was the severed body of a dead rat. The rear end was put on display somewhere together with the sign: "Who gives a rat's arse? Met do!"