Above, in a comment, Hao Sun asked if it was true that $\mathcal{H}om_X(\omega, \omega) = \mathcal{O}_X$. Here I will assume that $X$ is reduced and irreducible and $\omega$ is the canonical sheaf as discussed in S'andor's answer.
This second question is true if and only if the variety in question is S2 (so in particular, it holds in the Cohen-Macaulay case).
In fact, $\mathcal{H}om_X(\omega, \omega)$ is the S2-ification of $\mathcal{O}_X$ (and thus, Spec of it is the S2-ification of $X$).
EDIT: A reference for this last fact is Aoyama, "Some basic results on canonical modules", also see Aoyama, "On the depth and projective dimension of the canonical module".
EDIT2: Let me also sketch an idea for why this last statement is true. On a non-S2 variety $X$, the canonical sheaf of $X$ is the same as the canonical sheaf of the S2-ification of $X$ (up to pushdown). To see this, observe that S2-ification is an operation outside a set of codimension 2, and also observe that $\omega$ itself is always an S2-sheaf for a variety (see for example, a paper of Hartshorne, "Generalized divisors and biliaison"). It then quickly follows that $$ \mathcal{H}om_{O_X}(\omega,\omega) = \mathcal{H}om_{O_{X^{S2}} }(\omega, \omega).$$
This sheaf is clearly S2 (since $\omega$ is), furthermore, $$R\mathcal{H}om_{O_{X^{S2}} }^{.}(\omega^{.}, \omega^{.}) \cong {\mathcal O_{X^{S2}}}.$$, all the other terms appearing in that spectral sequences used to compute that have support at a codimension-2 subset, and it follows that $\mathcal{O}_{X^{S2}}$ and $\mathcal{H}om_{O_{X^{S2}} }(\omega, \omega)$ are isomorphic outside a set of codimension 2, and the result follows.