Suppose $f: X \rightarrow Y$ is a finite, flat (hence locally free) morphism of curves (i.e. schemes of dimension 1, not smooth or even reduced). Suppose $L$ is a reflexive sheaf on $X$, locally free of rank 1 at each generic point of $X$.
Is the direct image $f_* L$ still reflexive on $Y$? (Better its top exterior power). What if $Y$ is smooth?
Here is a (probably non-optimal) statement that may apply in your situation. In your situation with curves, the hypothesis says that you need $X$ and $Y$ to be Gorenstein.
We start by defining the $G_r$ condition.
Definition [Reiten and Fossum 1970, p. 142]. Let $X$ be a locally Noetherian scheme and fix an integer $r \ge 0$. We say that $X$ satisfies $G_r$ if $\mathcal{O}_{X,x}$ is Gorenstein for every point $x \in X$ such that $\dim(\mathcal{O}_{X,x}) \le r$.
Claim. Let $X$ and $Y$ be Noetherian schemes satisfying $G_1$ and $S_2$. Suppose that $f\colon X \to Y$ is a finite surjective morphism. If $\mathscr{F}$ is a coherent reflexive sheaf on $X$, then $f_*\mathscr{F}$ is a coherent reflexive sheaf on $Y$.
Proof. On Noetherian schemes satisfying $G_1$ and $S_2$, reflexivity is equivalent to being $S_2$ (in Samuel's [1964, Proposition 6] sense [Vasconcelos 1968, Definition 1.1]) by [Vasconcelos 1968, Theorem 1.4; Hartshorne 1994, Theorem 1.9]. The claim then follows since the $S_r$ property is preserved under pushforward by finite surjective morphisms by [EGAIV$_2$, Proposition 5.7.9]. $\blacksquare$
I wanted to prove a statement for non-finite morphisms as well, and for integral schemes, you can say a bit more:
Claim. Let $X$ and $Y$ be integral Noetherian schemes satisfying $G_1$ and $S_2$. Suppose that $f\colon X \to Y$ is a proper dominant morphism with all fibers of the same dimension. If $\mathscr{F}$ is a coherent reflexive sheaf on $X$, then $f_*\mathscr{F}$ is a coherent reflexive sheaf on $Y$.
Proof. The fact that $f_*\mathscr{F}$ is coherent and normal follows from the proof of [Hartshorne 1980, Corollary 1.7]. By [Hartshorne 1994, Remark 1.11], to show $f_*\mathscr{F}$ is reflexive, it therefore suffices to show that it satisfies $S_1$. But being $S_1$ is equivalent to torsion-freeness for integral noetherian schemes [Hartshorne 1994, Lemma 1.5], hence the claim follows by the fact that torsion-freeness is preserved under pushforwards by dominant morphisms. $\blacksquare$
Remark. There are other notions called $G_r$ in the literature. For example, $S_r$ + Reiten and Fossum's condition $G_r$ is what Marinari calls $G_r$ in [Marinari 1972, Definition 4.5]. The condition $S_{r+1} + G_r$ is what Ischebeck calls $G_r$ in [Ischebeck 1969, Definition 3.16].
Edit 1. Added the hypothesis that $f$ is surjective in the first claim.
Edit 2. To address Shrugs's request in the comments, I added the definition of $G_r$ with some references to other definitions in the literature.
