This involves Memory of Light
I just finished the last book in WoT and I don't recall a moment where the Tinkers find the song that they are searching for. I recall that the Ogier sing at one point but that is about it. Did I miss something?
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No, the Tinkers will not find their song.
Robert Jordan's notes on this are very clear: the Tinkers will never find their song. They've lost it for too long, that even if someone stood in front of them singing The Song, they would just nod their head, say 'that's a nice song' and go on their way.
From an Interview transcript on Theoryland
The Song itself is the growing song Rand heard in Rhuidean.
The Song the Tinkers are seeking is the song Rand heard in Rhuidean—or, to be exact, the memories of that song and others like it have become merged, over the years, into the concept of one mystical Song.
From another Interview transcript on Theoryland.