The “subject + ne + verb + pas” is the correct way to say it, but the “subject + verb + pas” is the informal way to say it.

Example :

I am not a boy

Will be translated in a formal way by

Je ne suis pas un garçon

And in an informal way

Je suis pas un garçon

French people tend to choose the informal one in everyday life, but if you study French, I’ll advise you to use the formal version.

In addition, in those lyrics of Stromae, he’s playing a drunk person (falling, barely stay on his foot etc..) which justifies the use of the second version.

It is really common to remove ne in spoken language, particularly in informal talks, with your friends or family for example, even though it is not grammatically correct.

For written language you can also skip it in informal talks, but again, it’s not grammatically correct.
In formal written language you should always use ne

Actually, it is a very good question. Most native speakers don’t use ne when they speak but use it when they write. I do that too.

However, as stated above, if you study french, the best way is to use ne when you write so you can get a good habit in writing French. After being used to that for a while, you may remove the ne when you speak, as it’s more French and you’ll sound more like a native.