Let me expand Donu's answer a bit. Say that $Y$ is normal for simplicity. Then we always have:
$$f_* \omega_X \subseteq \omega_Y.$$
The sheaf $\omega_Y$ is Cohen-Macaulay if $Y$ is Cohen-Macaulay, but in general, it is not.
On the other hand, $\omega_Y$ is always S2 (see for instance Kollár-Mori). Furthermore, if $Y$ is normal, then $f_* \omega_X \subseteq \omega_Y$ is an isomorphism outside a set of codimension 2. Hence
Observation For normal $Y$, $f_* \omega_X$ will be S2 if and only if $f_* \omega_X = \omega_Y$. This equality is equivalent to rational singularities if $Y$ is Cohen-Macaulay.
In other words, the only reasonable way to guarantee that $f_* \omega_X$ is Cohen-Macaulay is to require that $Y$ has rational singularities (at least if $Y$ is normal).