For a discrete group G, the kernel of a homomorphism from bounded cohomology $hat{H}^*(G)$ of G to the ordinary cohomology $H^*(G)$ of G is called the singular part of $hat{H}^*(G)$. We give some results on the space of the singular part of the second bounded cohomology of G. Also some results on the second bounded cohomology of a uniformly perfect group are given.