  The singular see-saw mechanism is a variation of the see-saw mechanism
whereby the right-chiral neutrino Majorana mass matrix is singular. Previous
works employing the singular see-saw mechanism have assumed a vanishing
left-chiral Majorana mass matrix. We study the neutrino spectrum obtained under
a singular see-saw mechanism when the left-chiral neutrinos possess a non-zero
Majorana mass matrix. We refer to this as the type-II singular see-saw
mechanism. The resulting neutrino spectrum is found to be sensitive to the
hierarchy of the Dirac and Majorana mass scales used and we explore the
phenomenological consequences of the candidate hierarchies. The compatibility
of the resulting spectra with the body of neutrino oscillation data is
discussed. It is found that neutrino mass matrices with this structure result
in 3+1 or 2+2 neutrino spectra, making it unlikely that this mass matrix
structure is realized in nature. If the left-chiral Majorana mass matrix is
also singular we show that a type-II singular see-saw mechanism can realize a
spectrum of one active-sterile pseudo-Dirac neutrino in conjunction with two
active Majorana neutrinos effectively decoupled from the sterile sector. This
realizes a scheme discussed in the literature in relation to astrophysical
neutrino fluxes.
