Rat’s arse

This is a modern form of Tinker’s cuss and dates from the late 20th century. The expression, not to give a rat’s arse (ass) means that one could not care less. In idiomatic speech down the years, animals of all kinds come and go according to whim or fashion. One would imagine that the arse or ass of a rat is fairly small and insignificant compared to some animals, hence the expression.