English has a collection of negative-polarity idiom frames with minimal direct objects involving an assortment of conventionalized nouns. These are often cited (in idiom and slang dictionaries, for example) with an explicit negator not:
(1a) not give a ___
(1b) not care a ___
(2) not be worth a ___
Fillers include (but are not limited to) damn, fuck, shit, crap; hang, hoot, lick, whit, rip, fig; some expanded nominals (flying fuck, good goddamn); and some longer nominals (hill of beans, rat’s ass / arse). A few mass nouns can serve as fillers: not be worth fuck / shit / crap. There is considerable variation here, from idiom to idiom and person to person, as to which nouns can fill which slots, and the patterns are sometimes playfully or euphemistically extended to new items: not give a turd, not give a doggone.
The fillers for the slots are mostly nouns referring to ’small, valueless, or contemptible’ things (as the relevant subentry for fig in the OED puts it), but there seem to have been plenty of accidents of history in way particular nouns were conventionalized in these slots.
Pattern (2) is straightforward syntactically; it’s just a predicative construction. But patterns (1a) and (1b) have a fairly complex external syntax, occurring with no complement (with the object of scorn supplied from context), with the object of scorn expressed by a PP (most often in about) or in a clause (a WH interrogative clause, an if clause, or a that clause):
Frankly, I don’t give a damn.
I don’t give a damn about your feelings.
I don’t give a damn who you are / what you do / when you go.
I don’t give a damn if / that you can speak Swahili.
None of these observations are especially new, certainly not original with me. My interest here is in the origin of some of the more remarkable fillers, in particular rat’s ass / arse.
(For some discussion of a tinker’s damn, or dam, see Michael Quinion’s entry here.)
Now, a rat’s ass is small, worthless, and contemptible, so rat’s ass is an eligible filler, and in fact it occurs (with considerable frequency) in all three patterns:
Some of us don’t give a rat’s ass about schools, you know. Take it to the school board! (link)
Does he really want to lose the independents over this crap, that he does not care a rat’s ass about. (link)
I got into an argument the other day, and this guy says my opinion “isn’t worth a rat’s ass.” Now, I’m no genius, but does he know the true value of a rat’s ass? (link)
Rat’s ass is not an expression that people would be likely to pull up out of the air; it looks like a playful creation on the part of some individual speaker, complete with assonance (in American English, anyway), which then spread to others. The innovator wouldn’t have had to be a public figure of any sort, though its use by such a figure would have hastened its spread.
So rat’s ass is something of a trial for lexicographers (once again, as here, I note that I’m not a lexicographer, and I appreciate how difficult lexicography is): determined digging in texts will give us a lower bound on how long it’s been around, but slang like rat’s ass takes some time to make it into print (often appearing first in reports of speech), so it might be much older than texts would suggest.
On-line resources are not very helpful, though they can be entertaining. One dictionary of intenet slang lists idgara, along with idgac (crap), idgad (damn), idgaf (fuck), idgaff (flying fuck), idgafs (fucking shit), and idgas (shit). And a (non-scholarly) dictionary of Australian slang lists rat’s ass as a piece of “Aussie lingo”.