Both are common. One is spoken as a fact and the other is spoken sarcastically, but the meaning is the same.

I agree with what others have said about how these sentences are used. But their literal meanings are opposite. The first literally means “It WOULD NOT be possible for me to care less than I do,” that is, I do not care at all. The literal meaning of the second sentence is, “It WOULD be possible for me to care less than I do,” that is, my not caring is not very strong. The literal meaning of the first sentence is what people mean when they use either sentence. In other words, the second sentence does not literally mean what people use it to mean.

As an English speaker, I can tell you that both of these phrases are used and mean approximately the same thing, but the tone with which they are said makes all the difference. In writing, you will most likely only see the first phrase used. The second phrase is also used when speaking, but in a sarcastic way. As another person has noted, both can be perceived as rude depending on the context, tone, and setting.