I will describe some new computations of the second bounded cohomology (with trivial real coefficients) of a large class of groups of homeomorphisms of 1-manifolds, including groups of PL and piecewise projective homeomorphisms. I will also discuss applications of this to some problems concerning the spectrum of stable commutator length of finitely presented groups and simplicial volume of manifolds. This is joint work with Francesco Fournier-Facio.