  In the past few years, coupling strengths between light and mechanical motion
in optomechanical setups have improved by orders of magnitude. Here we show
that, in the standard setup under continuous laser illumination, the steady
state of the mechanical oscillator can develop a non-classical, strongly
negative Wigner density if the optomechanical coupling is large at the
single-photon level. Because of its robustness, such a Wigner density can be
mapped using optical homodyne tomography. These features are observed near the
onset of the instability towards self-induced oscillations. We show that there
are also distinct signatures in the photon-photon correlation function
$g^{(2)}(t)$ in that regime, including oscillations decaying on a time scale
not only much longer than the optical cavity decay time, but even longer than
the \emph{mechanical} decay time.
