  The discovery of a Higgs boson with a mass of 126 GeV at the LHC when
combined with the non-observation of new physics both in direct and indirect
searches imposes strong constraints on supersymmetric models and in particular
on the top squark sector. The experiments for direct detection of dark matter
have provided with yet more constraints on the neutralino LSP mass and its
interactions. After imposing limits from the Higgs, flavour and dark matter
sectors, we examine the feasibility for a light stop in the context of the
pMSSM, in light of current results for stop and other SUSY searches at the LHC.
We only require that the neutralino dark matter explains a fraction of the
cosmologically measured dark matter abundance. We find that a stop with mass
below $\sim$ 500 GeV is still allowed. We further study various probes of the
light stop scenario that could be performed at the LHC Run - II either through
direct searches for the light and heavy stop, or SUSY searches not currently
available in simplified model results. Moreover we study the characteristics of
heavy Higgs for the points in the parameter space allowed by all the available
constraints and illustrate the region with large cross sections to fermionic or
electroweakino channels. Finally we show that nearly all scenarios with a small
stop$-$LSP mass difference will be tested by Xenon1T provided the NLSP is a
chargino, thus probing a region hard to access at the LHC.
