Another way to read the effect of the repitition of the word "singing" is as a "democratizing effect". By associating various types of people with the same musical quality, Whitman suggests that all these types of people (or all people, generally) are possessed of the same essential beauty, musicality, and soul.

The word I notice repeated most often is "singing," which ties in with Whitman's title, "I Hear America Singing."

The sense that all people sing has the effect of joining diverse groups together with song in common. It is, after all, a universal language.

I believe the use of repetition does three things in this poem: it shows that people from all walks of life sing their own song—it is a part of them and it shows a sense of pleasure in what they do or in their circumstances, in general.

The repetition may also create a rhythmic sense of movement, as the songs move back and forth between so many different kinds of people.

Of course, repetition is also used by an writer to stress something especially important to him or her:  it must be singing.

There are a couple words I think are key.  They are "singing" and "I."

I think "singing" is probably the one you want.  To me, the repetition makes it sound something like the chorus of a song.  This adds to the idea of America singing.