Yes, 'ain't' is used in most English-speaking countries, especially the US and the UK. It is a lazy person's negative, and can replace a number of different forms, most usually 'am not', 'is not', 'are not', 'have not, and 'has not'.
But, as Jean says, it is not generally seen as correct English, and the use of 'ain't' is widely associated in many people's minds with sections of society who do not have a good level of education.
Note that popular songs often contain non-standard language, such as 'he don't'. There are two main reasons for this:
1) Non-standard language, such as 'he don't' and 'I ain't', is often shorter and easier to say. This means that it fits better into the lyrics and goes better with the music than standard English does.
2) Popular music aims attract to a wide and mainly young audience. Songwriters want the songs to sound 'cool' and attractive to young people. The use of language which listeners' teachers and parents disapprove of increases this appeal.
By the way, it's worth bearing in mind that Adele, in common with many British artists, uses American English for her writing and singing. What you hear her sing is very different from the English she uses in her daily life.
Should you use 'ain't'? No. This is one of the many aspects of English of which you need only a 'passive knowledge'. You need to recognise and understand people when you hear them say things such as 'I ain't done', 'He don't know', 'We got a problem' , and 'I don't know nothing', but you should never use these non-standard forms yourself.
Listen, but don't repeat.