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The observation of charged lepton flavour violation would be a smoking gunfor new physics and could help in pinpointing the mechanism at the origin ofneutrino masses and mixing. We present here our recent studies of leptonflavour violating Higgs decays in the inverse seesaw and its supersymmetricembedding, two examples of low-scale seesaw mechanisms. We predict branchingratios 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 futurecolliders. Supersymmetric contributions can enhance the branching ratio of$h\rightarrow \tau \mu$ up to $1\%$, making it large enough to explain thesmall excess observed by ATLAS and CMS.

Lepton Flavour Violating Higgs Decays in the (SUSY) Inverse Seesaw