  We perform the Hamiltonian analysis of several mimetic gravity models and
compare our results with those obtained previously by different authors. We
verify that for healthy mimetic scalar-tensor theories the condition for the
corresponding part of the Hamiltonian to be bounded from below is the positive
value of the mimetic field energy density $\lambda$. We show that for mimetic
dark matter possessing a shift symmetry the mimetic energy density remains
positive in time, provided appropriate boundary conditions are imposed on its
initial value, while in models without shift symmetry the positive energy
density can be maintained by simply replacing $\lambda\to e^{\lambda}$. The
same result also applies to mimetic $f(R)$ gravity, which is healthy if the
usual stability conditions of the standard $f(R)$ gravity are assumed and
$\lambda>0$. In contrast, if we add mimetic matter to an unhealthy seed action,
the resulting mimetic gravity theory remains, in general, unstable. As an
example, we consider a scalar-tensor theory with the higher-derivative term
$(\Box \varphi)^2$, which contains an Ostrogradski ghost. We also revisit
results regarding stability issues of linear perturbations around the FLRW
background of the mimetic dark matter in the presence of ordinary scalar
matter. We find that the presence of conventional matter does not revive
dynamical ghost modes (at least in the UV limit). The modes, whose Hamiltonian
is not positive definite, are non-propagating (have zero sound speed) and are
associated with the mimetic matter itself. They are already present in the case
in which the ordinary scalar fluid is absent, causing a growth of dust
overdensity.
