  We investigate the interactions of a light scalar with the Higgs boson and
second-generation fermions, which trigger new rare decays of the Higgs boson
into $4\mu$, $2\mu 2\gamma$, $6\mu$ and $4\mu 2j$. We recast current LHC
searches to constrain these decays and develop new collider analyses for those
channels which are only poorly tested by existing studies. With the currently
collected data we can probe branching ratios as small as $1.5 \times10^{-5}$,
$8.7 \times10^{-5}$, $5.7 \times10^{-8}$ and $1.6 \times10^{-7}$, respectively.
For the High-Luminosity LHC run, considered here to involve 3 ab$^{-1}$ of
integrated luminosity, these numbers go down to $1.3 \times10^{-5}$, $2.0
\times10^{-6}$, $3.0 \times10^{-9}$ and $5.4 \times10^{-9}$, respectively. We
also comment on other channels that remain still unexplored.
