  We investigate the topological phases that appear in an orbital version of
the Benalcazar-Bernevig-Hughes (BBH) model in the presence of conventional
spin-singlet $s$-wave superconductivity and with the possibility of tuning an
in-plane magnetic field. We chart out the phase diagram by considering
different boundary conditions, with the topology of the individual phases
further examined by considering both the Wannier and entanglement spectra, as
well as the Majorana polarization.
  For weak to moderate values of magnetic field and superconducting pairing
amplitude, we find a second-order topological superconductor phase with eight
zero-energy corner modes. Further increasing field or pairing, half of the
corner states can be turned into zero-energy edge-localized modes, thus forming
a type of hybrid-order phase. Then, we find two different putative first-order
topological phases, a nodal and a nodeless phase, both with zero-energy flat
bands localized along mirror-symmetric open edges. For the nodal phase, the
flat bands are localized between the nodes in reciprocal space, while in the
nodeless phase, with its a full bulk gap, the zero-energy boundary flat band
spans the whole Brillouin zone.
