  We study the representation theory of the N=1 super Heisenberg-Virasoro
vertex algebra at level zero, which extends the previous work on the
Heisenberg-Virasoro vertex algebra arXiv:math/0201314, arXiv:1405.1707 and
arXiv:1703.00531 to the super case. We calculated all characters of irreducible
highest weight representations by investigating certain Fock space
representations. Quite surprisingly, we found that the maximal submodules of
certain Verma modules are generated by subsingular vectors. The formulas for
singular and subsingular vectors are obtained using screening operators
appearing in a study of certain logarithmic vertex algebras in arXiv:0908.4053.
