  We carry out a systematic investigation for the minimal Dirac neutrino mass
models emerging from generic one-loop and two-loop topologies that arise from
$d=5$ effective operator with a singlet scalar, $\sigma$. To ensure that the
tree-level Dirac mass, as well as Majorana mass terms at all orders, are absent
for the neutrinos, we work in the framework where the Standard Model is
supplemented by the well-motivated $U(1)_{B-L}$ gauge symmetry. At the one-loop
level, we analyze six possible topologies, out of which two of them have the
potential to generate desired Dirac neutrino mass. Adopting a systematic
approach to select minimal models, we construct seventeen viable one-loop Dirac
neutrino mass models. By embracing a similar methodical approach at the
two-loop, we work out twenty-three minimal candidates. Among the forty selected
economical models, the majority of the models proposed in this work are new. In
our search, we also include the scenarios where the particles in the loop carry
charges under the color group. Furthermore, we discuss the possible dark matter
candidates within a given model, if any, without extending the minimal particle
content.
