My informant is my mother, who has heard my father spout says and folk speech all their married life. My father is prone to using such phrases in everyday conversations. Here is an example she gives when I asked if she knew any folk sayings.
“Your father says “I don’t give a rat’s ass” all the time, usually when you are complaining to him about something, he’ll say it and you will know the argument is pointless because he doesn’t care about the circumstances or what you are saying, he’s just going to continue on what he is doing because he thinks it doesn’t apply to them. Like if you were to complain about . . . say doing a chore or homework or something and give him some excuses he would say he doesn’t give a rat’s ass and tell you to go do whatever it is anyway”
This basically means that the person who uses this folk speech doesn’t care about the circumstances whatsoever, even to the extent of a rat’s ass. No one cares much about rats in fact most people hate them, and their “ass” is even more insignificant than the whole rat, thus creating the effect of making an allusion to the least most desirable thing imaginable. Thus if the circumstance is cared about less than a rat’s ass, then they really do not care. It is usually used amongst friends or family as it is quite explicit and either in joking or a serious argument.