Which is correct: "could care less" or "couldn't care less"?
Solution 1:
I've heard it said that "could care less" is meant to be ironic, but I think this is just justification for the bastardisation of an English phrase.
Here we go (from World Wide Words):
There’s a close link between the stress pattern of I could care less and the kind that appears in certain sarcastic or self-deprecatory phrases that are associated with the Yiddish heritage and (especially) New York Jewish speech. Perhaps the best known is I should be so lucky!, in which the real sense is often “I have no hope of being so lucky”, a closely similar stress pattern with the same sarcastic inversion of meaning. There’s no evidence to suggest that I could care less came directly from Yiddish, but the similarity is suggestive. There are other American expressions that have a similar sarcastic inversion of apparent sense, such as Tell me about it!, which usually means “Don’t tell me about it, because I know all about it already”. These may come from similar sources.
Solution 2:
I also think "couldn't care less" is the correct form since it is more logical. Perhaps this is a meta issue in that precisely those people who say "could care less" could not indeed care less whether they are speaking logically or not.
Solution 3:
You should feel free to say either variety. "Could care less" actually occurs more frequently. It is an entrenched idiom. No fluent speaker will have any trouble understanding what you mean. The Oxford English Dictionary lists both with the same meaning:
care
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(4). In negative and conditional construction: a. not to care passes from the notion of ‘not to trouble oneself’, to those of ‘not to mind, not to regard or pay any deference or attention, to pay no respect, be indifferent’.
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(c) Colloq. phr. (I, etc.) couldn't care less: (I am, etc.) completely uninterested, utterly indifferent; freq. as phr. used attrib. Hence couldn't-care-less-ness.
(d) U.S. colloq. phr. (I, etc.) could care less = sense (c) above, with omission of negative.
If you're interested in a linguist's thoughts on the topic, you can start reading this article, and continue with all the linked articles at the bottom of that.
That being said, there are two reasons you might want to avoid "could care less":
There are a lot of people that peeve about "could care less," and if you're worried about offending them maybe you shouldn't use the phrase. (On the other hand, now that you've read this answer and the articles I've linked to, you can confidently tell them that they're actually the ones who are wrong, and "could care less" is perfectly fine, and if they have a problem with it then it's because they are trying to alter the English language.)
When speaking to non-fluent speakers, you should attempt to avoid idioms like "could care less" or "raining cats and dogs". If they aren't familiar with the idiom, they might try to parse it literally.
Solution 4:
Originally the expression was "I couldn't care less" which, as you pointed out, makes sense. Over time, this was confused and turned into "I could care less" which, of course, doesn't make sense.