  The ATLAS and CMS experiments at the LHC have reported the observation of a
possible excess of events corresponding to a new particle $h$ with mass $\sim
125$ GeV that might be the long-sought Higgs boson, or something else.
Decyphering the nature of this possible signal will require constraining the
couplings of the $h$ and measuring them as accurately as possible. Here we
analyze the indirect constraints on flavour-changing $h$ decays that are
provided by limits on low-energy flavour-changing interactions. We find that
indirect limits in the quark sector impose such strong constraints that
flavour-changing $h$ decays to quark-antiquark pairs are unlikely to be
observable at the LHC. On the other hand, the upper limits on
lepton-flavour-changing decays are weaker, and the experimental signatures less
challenging. In particular, we find that either ${\mathcal B}(h \to \tau \bar
\mu + \bar \mu \tau)$ or ${\mathcal B}(h \to \tau \bar e + \bar e \tau) $ could
be ${\cal O}(10)%$, i.e., comparable to ${\mathcal B}(h \to \tau^+ \tau^-)$ and
potentially observable at the LHC.
