  We study the response of random singlet quantum critical points to local
perturbations. Despite being insulating, these systems are dramatically
affected by a local cut in the system, so that the overlap $G=\left|\langle
\Psi_B |\Psi_A \rangle\right|$ of the groundstate wave functions with and
without a cut vanishes algebraically in the thermodynamic limit. We analyze
this Anderson orthogonality catastrophe in detail using a real-space
renormalization group approach. We show that both the typical value of the
overlap G and the disorder average of $G^\alpha$ with $\alpha>0$ decay as
power-laws of the system size. In particular, the disorder average of
$G^\alpha$ shows a "multifractal" behavior, with a non-trivial limit $\alpha
\to \infty$ that is dominated by rare events. We also discuss the case of more
generic local perturbations and generalize these results to local quantum
quenches.
