Is the symmetric product of an abelian variety a CY variety?

When $\dim A = 1$, $S^nA$ is a $\mathbb{P}^{n-1}$-bundle over $A$, so its Kodaira dimension is $-\infty$.

When $\dim A = 2$, the minimal resolution of $S^nA$ is given by the Hilbert scheme $A^{[n]}$, there is a natural map $$ A^{[n]} \to A $$ (summation of points), which is smooth with fiber $K_{n-1}A$, so-called higher Kummer variety, which is hyperkahler.

You need for the dimension of $A$ to be even in order for the canonical sheaf on the quotient to be trivial (so that the resolution has a chance to be $K$-trivial). For $\dim A =2$, the story is as Sasha described. For $\dim A $ even and at least 4, there will not exist a crepant resolution. You will encounter singularities which locally look like $\mathbb{C}^{2d}/\{\pm 1\}$ and for $d>1$ these do not admit crepant resolutions.