Those who "sing" in Whitman's poems are those who are the defining elements of the basis in America. Each particular part of America has a song to sing. In this praising of America's heterogeneous nature, each song is uniquely different from one another. However, Whitman ensures that each voice from a different walk of life is one that will be able to contribute to the tapestry of what it means to be America. The mechanics, the boatman, the women all provide distinct and unique voices in what it means to be America. These voices are middle class, or regular people, whose contributions define in Whitman's mind what it means to be American and the greatness within that essence.
Whitman tells us exactly who the singers are. He lists them out. There are mechanics and masons, carpenters and boatmen. There are men and women -- we see women sewing and washing. All of them are singing and their songs are uniquely their own.
All of these people, in my opinion, are not really important for their specific professions. In other words, it doesn't matter if Whitman has them being masons or fishermen. What's important here is that they are all common Americans. Whitman is saying that it is the common people who make America what it is.