  The observation of charged lepton flavour violation would be a smoking gun
for new physics and could help in pinpointing the mechanism at the origin of
neutrino masses and mixing. We present here our recent studies of lepton
flavour violating Higgs decays in the inverse seesaw and its supersymmetric
embedding, two examples of low-scale seesaw mechanisms. We predict branching
ratios as large as $10^{-5}$ for the decays $h\rightarrow \tau \mu$ and $h
\rightarrow \tau e$ in the inverse seesaw, which can be probed in future
colliders. Supersymmetric contributions can enhance the branching ratio of
$h\rightarrow \tau \mu$ up to $1\%$, making it large enough to explain the
small excess observed by ATLAS and CMS.
