  Mapping class groups satisfy cohomological stability. In this note we show
how results by Bestvina and Fujiwara imply that the bounded cohomology does not
stabilize, additionally we show that stabily polynomials in the
Mumford-Morita-Miller classes are unbounded i.e. their norm tends to infinity
as one increases the genus. While the bounded cohomology of the symplectic
group does stabilize, we show that it does not stabilize via isometries in
degree $2$. In order to establish this we calculate the norm of the signature
class in $\mathrm{Sp}_{2h}(\mathbb{R})$ and estimate the norm of the integral
signature class.
