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Just for the record, this does not need CM. The following is true:

Let $Z$ be an excellent scheme that admits a dualizing complex. Then $ω_Z$ is torsion-free and $S_2$ on $Z$. If in addition Z is irreducible, then $ω_Z$ is a reflexive $\mathscr O_Z$ -module.

If there is interest in this, I will add a proof.

As requested, here is a proof: (The following is Lemma 3.7.5 in this paper.)

Lemma Let $Z$ be an excellent scheme that admits a dualizing complex. Then $\omega_Z$ is torsion-free and $S_2$ on $Z$. If in addition $Z$ is normal, then $\omega_Z$ is a reflexive $\mathscr O_Z$-module.

Proof. The statement is local, so we may assume that $Z$ is a noetherian affine local scheme. Then, since it admits a dualizing complex, it can be embedded into a finite dimensional Gorenstein affine local scheme W as a closed subscheme by [Kaw02, Cor. 1.4]. Being Gorenstein and local, $W$ must be pure dimensional. Let $r = \mathrm{codim}(Z, W)$. Then by [Mat80, (16.B) Theorem 31(i)] there exists a length $r$ regular sequence in the ideal of $Z$ in $W$ and let $W'$ be the common zero locus. Then $W'$ is also Gorenstein, $Z\subseteq W$ is a closed subscheme and $\dim Z = \dim W$ .

It follows that $\omega_ {W'}^\bullet \simeq \omega_{W'}[m]$, where $\omega_{W'}$ is a line bundle on ${W'}$. By Grothendieck duality $\omega^\bullet_ Z\simeq RHom_{W'}(\mathscr O_Z, \omega^\bullet_{W'})$, and since $\dim Z=\dim W'$, $\omega_Z\simeq Hom_{W'}(\mathscr O_Z, \omega_{W'})$ hence it is indeed torsion-free on $Z$.

By [Stacks Project, Tag 0AWE] $\omega_Z$ is $S_2$. In particular, if $Z$ is normal, then it is reflexive by [Stacks Project,Tag 0AVB].

As far as non-trivial examples of schemes admitting dualizing complexes are concerned, try anything that can be embedded (locally) into a Gorenstein scheme. In fact, those are exactly the ones you are looking for.

Just for the record, this does not need CM. The following is true:

Let $Z$ be an excellent scheme that admits a dualizing complex. Then $ω_Z$ is torsion-free and $S_2$ on $Z$. If in addition Z is normal, then $ω_Z$ is a reflexive $\mathscr O_Z$ -module.

Also note that torsion-free and $S_2$ seems to be the "right" notion to replace reflexivity on non-normal schemes.

If there is interest in this, I will add a proof.

As requested, here is a proof: (The following is Lemma 3.7.5 in this paper.)

Lemma Let $Z$ be an excellent scheme that admits a dualizing complex. Then $\omega_Z$ is torsion-free and $S_2$ on $Z$. If in addition $Z$ is normal, then $\omega_Z$ is a reflexive $\mathscr O_Z$-module.

Proof. The statement is local, so we may assume that $Z$ is a noetherian affine local scheme. Then, since it admits a dualizing complex, it can be embedded into a finite dimensional Gorenstein affine local scheme W as a closed subscheme by [Kaw02, Cor. 1.4]. Being Gorenstein and local, $W$ must be pure dimensional. Let $r = \mathrm{codim}(Z, W)$. Then by [Mat80, (16.B) Theorem 31(i)] there exists a length $r$ regular sequence in the ideal of $Z$ in $W$ and let $W'$ be the common zero locus. Then $W'$ is also Gorenstein, $Z\subseteq W$ is a closed subscheme and $\dim Z = \dim W$ .

It follows that $\omega_ {W'}^\bullet \simeq \omega_{W'}[m]$, where $\omega_{W'}$ is a line bundle on ${W'}$. By Grothendieck duality $\omega^\bullet_ Z\simeq RHom_{W'}(\mathscr O_Z, \omega^\bullet_{W'})$, and since $\dim Z=\dim W'$, $\omega_Z\simeq Hom_{W'}(\mathscr O_Z, \omega_{W'})$ hence it is indeed torsion-free on $Z$.

By [Stacks Project, Tag 0AWE] $\omega_Z$ is $S_2$. In particular, if $Z$ is normal, then it is reflexive by [Stacks Project,Tag 0AVB].

As far as non-trivial examples of schemes admitting dualizing complexes are concerned, try anything that can be embedded (locally) into a Gorenstein scheme. In fact, those are exactly the ones you are looking for.

Just for the record, this does not need CM. The following is true:

Let $Z$ be an excellent scheme that admits a dualizing complex. Then $ω_Z$ is torsion-free and $S_2$ on $Z$. If in addition Z is irreducible, then $ω_Z$ is a reflexive $\mathscr O_Z$ -module.

If there is interest in this, I will add a proof.

As requested, here is a proof: (The following is Lemma 3.7.5 in this paper.)

Lemma Let $Z$ be an excellent scheme that admits a dualizing complex. Then $\omega_Z$ is torsion-free and $S_2$ on $Z$. If in addition $Z$ is normal, then $\omega_Z$ is a reflexive $\mathscr O_Z$-module.

Proof. The statement is local, so we may assume that $Z$ is a noetherian affine local scheme. Then, since it admits a dualizing complex, it can be embedded into a finite dimensional Gorenstein affine local scheme W as a closed subscheme by [Kaw02, Cor. 1.4]. Being Gorenstein and local, $W$ must be pure dimensional. Let $r = \mathrm{codim}(Z, W)$. Then by [Mat80, (16.B) Theorem 31(i)] there exists a length $r$ regular sequence in the ideal of $Z$ in $W$ and let $W'$ be the common zero locus. Then $W'$ is also Gorenstein, $Z\subseteq W$ is a closed subscheme and $\dim Z = \dim W$ .

It follows that $\omega_ {W'}^\bullet \simeq \omega_{W'}[m]$, where $\omega_{W'}$ is a line bundle on ${W'}$. By Grothendieck duality $\omega^\bullet_ Z\simeq RHom_{W'}(\mathscr O_Z, \omega^\bullet_{W'})$, and since $\dim Z=\dim W'$, $\omega_Z\simeq Hom_{W'}(\mathscr O_Z, \omega_{W'})$ hence it is indeed torsion-free on $Z$.

By [Stacks Project, Tag 0AWE] $\omega_Z$ is $S_2$. In particular, if $Z$ is normal, then it is reflexive by [Stacks Project,Tag 0AVB].

Just for the record, this does not need CM. The following is true:

Let $Z$ be an excellent scheme that admits a dualizing complex. Then $ω_Z$ is torsion-free and $S_2$ on $Z$. If in addition Z is irreducible, then $ω_Z$ is a reflexive $\mathscr O_Z$ -module.

If there is interest in this, I will add a proof.

As requested, here is a proof: (The following is Lemma 3.7.5 in this paper.)

Lemma Let $Z$ be an excellent scheme that admits a dualizing complex. Then $\omega_Z$ is torsion-free and $S_2$ on $Z$. If in addition $Z$ is normal, then $\omega_Z$ is a reflexive $\mathscr O_Z$-module.

Proof. The statement is local, so we may assume that $Z$ is a noetherian affine local scheme. Then, since it admits a dualizing complex, it can be embedded into a finite dimensional Gorenstein affine local scheme W as a closed subscheme by [Kaw02, Cor. 1.4]. Being Gorenstein and local, $W$ must be pure dimensional. Let $r = \mathrm{codim}(Z, W)$. Then by [Mat80, (16.B) Theorem 31(i)] there exists a length $r$ regular sequence in the ideal of $Z$ in $W$ and let $W'$ be the common zero locus. Then $W'$ is also Gorenstein, $Z\subseteq W$ is a closed subscheme and $\dim Z = \dim W$ .

It follows that $\omega_ {W'}^\bullet \simeq \omega_{W'}[m]$, where $\omega_{W'}$ is a line bundle on ${W'}$. By Grothendieck duality $\omega^\bullet_ Z\simeq RHom_{W'}(\mathscr O_Z, \omega^\bullet_{W'})$, and since $\dim Z=\dim W'$, $\omega_Z\simeq Hom_{W'}(\mathscr O_Z, \omega_{W'})$ hence it is indeed torsion-free on $Z$.

By [Stacks Project, Tag 0AWE] $\omega_Z$ is $S_2$. In particular, if $Z$ is normal, then it is reflexive by [Stacks Project,Tag 0AVB].

As far as non-trivial examples of schemes admitting dualizing complexes are concerned, try anything that can be embedded (locally) into a Gorenstein scheme. In fact, those are exactly the ones you are looking for.

Just for the record, this does not need CM. The following is true:

Let $Z$ be an excellent scheme that admits a dualizing complex. Then $ω_Z$ is torsion-free and $S_2$ on $Z$. If in addition Z is irreducible, then $ω_Z$ is a reflexive $\mathscr O_Z$ -module.

If there is interest in this, I will add a proof.

Just for the record, this does not need CM. The following is true:

Let $Z$ be an excellent scheme that admits a dualizing complex. Then $ω_Z$ is torsion-free and $S_2$ on $Z$. If in addition Z is irreducible, then $ω_Z$ is a reflexive $\mathscr O_Z$ -module.

If there is interest in this, I will add a proof.

As requested, here is a proof: (The following is Lemma 3.7.5 in this paper.)

Lemma Let $Z$ be an excellent scheme that admits a dualizing complex. Then $\omega_Z$ is torsion-free and $S_2$ on $Z$. If in addition $Z$ is normal, then $\omega_Z$ is a reflexive $\mathscr O_Z$-module.

Proof. The statement is local, so we may assume that $Z$ is a noetherian affine local scheme. Then, since it admits a dualizing complex, it can be embedded into a finite dimensional Gorenstein affine local scheme W as a closed subscheme by [Kaw02, Cor. 1.4]. Being Gorenstein and local, $W$ must be pure dimensional. Let $r = \mathrm{codim}(Z, W)$. Then by [Mat80, (16.B) Theorem 31(i)] there exists a length $r$ regular sequence in the ideal of $Z$ in $W$ and let $W'$ be the common zero locus. Then $W'$ is also Gorenstein, $Z\subseteq W$ is a closed subscheme and $\dim Z = \dim W$ .

It follows that $\omega_ {W'}^\bullet \simeq \omega_{W'}[m]$, where $\omega_{W'}$ is a line bundle on ${W'}$. By Grothendieck duality $\omega^\bullet_ Z\simeq RHom_{W'}(\mathscr O_Z, \omega^\bullet_{W'})$, and since $\dim Z=\dim W'$, $\omega_Z\simeq Hom_{W'}(\mathscr O_Z, \omega_{W'})$ hence it is indeed torsion-free on $Z$.

By [Stacks Project, Tag 0AWE] $\omega_Z$ is $S_2$. In particular, if $Z$ is normal, then it is reflexive by [Stacks Project,Tag 0AVB].