If $X$ is a curve with a nodal singularity at $x$, it's referred to here and here that its dualising sheaf is $$\omega_X \ = \ \pi_*(\Omega_{X}(p_1+\cdots+p_n)').$$ Here, $\pi:X\to X'$ is the normalisation and $\Omega_{X}(p_1+\cdots+p_n)'$ are the forms $\theta$ which have at worst simple poles at $\pi^{-1}(x)=\{p_1,...,p_n\}$, satisfying $$\sum_i \text{Res}_{p_i}\theta\ = \ 0 .$$

- What is the proof that this is the dualising sheaf? I can't find a reference, even in the complex-analytic case.
- What happens when $X$ has other types of singularities (and what is the proof)? For instance, working complex-analytically, what if $X$ is locally $$(z-\alpha_1w)\cdots(z-\alpha_nw)\ = 0 \ ?$$

Algebraic Groups and Class Fieldsby Serre is devoted to differentials on singular curves. This is pre-Grothendieck, so the term "dualizing sheaf" does not appear, but Serre proves that his sheaf of differentials is indeed a dualizing sheaf. $\endgroup$Moduli of Curves. I believe the proof is outlined there in the complex analytic case, as well as some discussion of singularities that cause this sheaf to no longer be a line bundle. $\endgroup$