The only time we sang cadence was in the 'wop-wops' (the bush, ie, marching in the mountains) to keep us awake.
I don't know about the Army, but in the airforce atleast it wasn't allowed (or considered necessary - in the words of my boot camp Corporal :'only the yanks sing when they march, because they can't count!' )
What we sang in the bush was:
Bee-bop-a-reeba (bee-bop-a-reeba)
Bee-bop-a-ro-o (repeated..)
I wish that all the ladies (...)
Were potholes in the road
And if I was a dump truck
I'd fill them with my load
A bee-bop-a-reeba ( repeat..)
A-bee-bop-a-ro-o (...)
I wish that all the ladies
were waves on the ocean
And if I was a surfer
I'd ride 'em with my motion (etc,etc..)
Unfortunately don't remember the other verses, but you get the idea...