  Motivated by the ATLAS and CMS discovery of a Higgs-like boson with a mass
around 125 GeV, and by the need of explaining neutrino masses, we analyse the
three canonical SUSY versions of the seesaw mechanism (type I, II and III) with
CMSSM boundary conditions. In type II and III cases, SUSY particles are lighter
than in the CMSSM (or the constrained type I seesaw), for the same set of input
parameters at the universality scale. Thus, to explain $m_{h^0} \simeq 125 GeV$
at low energies, one is forced into regions of parameter space with very large
values of $m_0$, $M_{1/2}$ or $A_0$. We compare the squark and gluino masses
allowed by the ATLAS and CMS ranges for $m_{h^0}$ (extracted from the 2011-2012
data), and discuss the possibility of distinguishing seesaw models in view of
future results on SUSY searches. In particular, we briefly comment on the
discovery potential of LHC upgrades, for squark/gluino mass ranges required by
present Higgs mass constraints. A discrimination between different seesaw
models cannot rely on the Higgs mass data alone, therefore we also take into
account the MEG upper limit on BR$(\mu \to e \gamma)$ and show that, in some
cases, this may help to restrict the SUSY parameter space, as well as to set
complementary limits on the seesaw scale.
