  We consider the contribution to the density of vibrational states and the
distribution of energy barrier heights of incipient instabilities in a glass
modeled by a jammed packing of spheres. On approaching an instability, the
frequency of a normal mode and the height of the energy barrier to cross into a
new ground state both vanish. These instabilities produce a contribution to the
density of vibrational states that scales as $\omega^3$ at low frequencies
$\omega$, but which vanishes in the thermodynamic limit. In addition, they
affect an anharmonic property, the distribution of energy barriers $\Delta H$,
giving a contribution that scales as $\Delta H^{-1/3}$ at low barrier heights,
which should be universal for jammed and glassy systems.
