"I can't get no satisfaction" The Rolling Stones
"Ain't no sunshine when she's gone" Bill Withers
You hit the nail on the head. Great artists can get away with breaking grammar rules. IELTS students can't!
Neither of those!
Gabriel, you have to understand that very few 'ordinary' educated English-speaking people would say either of those sentences. As we've already explained, both the word ain't and the use of double negatives are considered to be 'bad English'. They're OK in song lyrics, but not elsewhere.
In normal English, people say "I'm not telling you a lie". That's standard English grammar.
Alternatively, you could use a verb and say "I'm not lying to you" or "I'm telling you the truth".
Is it correct to say "I ain't telling you no lie"?
Yes. If a songwriter or scriptwriter is using the non-standard grammar of informal language and song lyrics, it is correct. Even non-standard grammar has its rules, and the rule with 'ain't' is that it takes a double negative. Similar examples are "I ain't going nowhere" or "I ain't saying nothing".
Should you use this kind of language?
No, you shouldn't. It's only appropriate to use 'ain't', double negatives, and other non-standard constructions such as she don't in certain kinds of popular music, or in movie dialogues to show that a character is ill-educated, for example. In normal, everyday English, you should use standard grammar: I'm not telling you a lie.
I think you mean informal conversation, not 'floppy'!
Ain't can actually replace many verb + not combinations : am not, is not, are not, have not and has not. But the distinction between ain't and standard forms is not one of formality. Particularly in the US, there is a great deal of stigma associated with this word, and it is viewed as a sign of poor education.
Native speakers occasionally say ain't as a kind of joke, when we're pretending to be ill-educated or when we're deliberately trying to sound 'folksy'. It's quite subtle, though, and I wouldn't recommend that a non-native try to use it. Just stick to normal contractions - I'm not and I don't.