  We study the multiplier ideals and the corresponding jumping numbers and
multiplicities $\{m(c)\}_{c\in \mathbb{R}}$ in the following context: $(X,o)$
is a complex analytic normal surface singularity, ${\mathfrak a}\subset
\mathcal{O}_{X,o}$ is an ${\mathfrak m}_{X,o}$--primary ideal,
$\phi:\widetilde{X}\to X$ is a log resolution of $\mathfrak{a}$ such that
$\mathfrak{a}\mathcal{O}_{\widetilde{X}}=\mathcal{O}_{\widetilde{X}}(-F)$, for
some nonzero effective divisor $F$ supported on $\phi^{-1}(0)$. We show that
$\{m(c)\}_{c>0}$ is combinatorially computable from $F$ and the resolution
graph $\Gamma$ of $\phi$, and we provide several formulae. We also extend
Budur's result (valid for $(X,o)=(\mathbb{C}^2,0)$), which makes an
identification of $\sum_{c\in[0,1]}m(c)t^c$ with a certain Hodge spectrum. In
our general case we use Hodge spectrum with coefficients in a mixed Hodge
module. We show that $\{m(c)\}_{c\leq 0}$ usually depends on the analytic type
of $(X,o)$. However, for some distinguished analytic types we determine it
concretely. E.g., when $(X,o)$ is weighted homogeneous (and $F$ is associated
with the central vertex), we recover $\sum_cm(c)t^c$ from the Poincar\'e series
of $(X,o)$ and when $(X,o)$ is a splice quotient then we recover
$\sum_cm(c)t^c$ from the multivariable topological Poincar\'e (zeta) function
of $\Gamma$.
