The Song of Growing were sang during a special event called Seed Singing during the Age of Legends to help plants growing fast and healthy. Da'shain Aiel sang the Song with Ogier together. It is unclear whether other townpeople sang it with them or not. Also unclear whether the Nym of the field was singing too, but the threads of the songs were braided together and the Nym wove them into the soil and seeds with his dance using with the One Power. 
In the Age of Legends people were provided with enough food. Aes Sedai were Servants of All, Da'shain Aiel were servants of Aes Sedai and humanity's needs were taken care of by them using the One Power and governing life.  Nyms were created for each field where plants were grown. At the event of Seed Singing the the owners of the fields were usually present. Da'shain Aiel and an Ogier knelt at the edge of the plowed ground surronded the field in a line with the others at twice-stretched-arm's length, all the way around. The Ogier began it, standing to sing, the Aiel rose, men's voice lifting in their own song, the Nym came and wove their threads into the soil and seeds through his dance gliding across the field. Where the Nym's foot had trod those plants had extra protection against insects and diseases and grow faster and bigger than others.
Seed Singing might have been held parallel so participants on one field could see others Singing on other fields and listening to their Singing.
Seed Singing first mentioned most probably in the Prologue of The Eye of the World when Lews Therin Telamon wanders among the ruins of his palace already mad from the taint of Saidin, looking for his wife Ilyena Sunhair and his children. Elan Morin Tedronai arrives to kill him or to turn him to the Dark One's side. Kinslayer does not recognise him in his madness, takes him as a stranger, a guest and asks him whether he has the Voice, having the time for the Singing and welcoming him to take part. It is not mentioned explicitly that he talks about Seed Singing, though.
There might be another indication here, that not everyone had sufficient voice for Seed Singing. When Rand watched the history of Aiel through the eyes of his ancestors in The Shadow Rising, one of his predecessors, Coumin was only sixteen but women decided his voice was finally deep enough to join in the seed singing.
References to Tuatha'an's missing Song
Sung by Rand al'Thor
Rand used the Song of Growing as a trick when he was shielded by a damane in Tuon's presence. He made them believe he is still using the One Power to make the grass green and make the trees blossom despite the shield. Mat hears him singing and the tune reminds him to a song, called "Two Maids at the Water's Edge". 
"Let the Prince of the Morning sing to the land that green things will grow." – The Cycle of the Dragon
Unexplained growings around Rand al'Thor
As Prophecy says "the land is one with the Dragon Reborn, and he is one with the land", Rand had unexplained effects on nature from the moment he won against the Shadow atop of Dragonmount. Some were related to his ta'veren status, as the Pattern balances the Shadow's wrongness through him such as the spoiled tea changed to an amazingly tasty one just by his holding the cup in hands for a few seconds. Others are happening purely by his presence. The apple trees at the bottom of Dragonmount have a complete cycle of growing leaves, blossoming and producting fruits just in a few minutes of time. 
—A Memory of Light
On the Field of Merrilor when Rand approached the place for the pavilion the grass greened at his feet. Where he trod, the turf recovered, spreading from him like a soft wave of light through opening shutters. Egwene wonders how he does it and when Gawyn explains it with the One Power, Egwene says it is too extensive for that. It feels so natural. 
During the discussion over the Dragon's Peace a grove of a dozen trees grows around the pavilion. Towering, healthy, huge-limbed, beautiful trees, hundreds of feet tall with massive trunks. Perrin states he has seen Great Trees like them before, inside a stedding. Shienaran state watching them grow, bursting from the ground, taking only five minutes to become full grown.  Although there was no mention about Rand singing, yet it is implied that he had done something. His reason is also not clarified, but he prepared for doing it by hanging thirteen glowing saidin globes of light at the corners.
During the breaking of the world, this song was forgotten even by Aiel and in time the search for it became the Tuatha`an's quest. [verify]