We consider a simple extension of the Standard Model that can account for the dark matter and explain the existence of neutrino masses. The model includes a vector-like doublet of SU(2), a singlet fermion, and two scalar singlets, all of them odd under a new Z$_2$ symmetry. Neutrino masses are generated radiatively by one-loop processes involving the new fields, while the dark matter candidate is the lightest neutral particle among them. We focus specifically on the case where the dark matter particle is one of the scalars and its relic density is determined by its Yukawa interactions. The phenomenology of this setup, including neutrino masses, dark matter and lepton flavor violation, is analyzed in some detail. We find that the dark matter mass must be below $500$ GeV to satisfy the relic density constraint. Lepton flavor violating processes are shown to provide the most promising way to test this scenario. Future $\mu\to 3e$ and $\mu$-$e$ conversion experiments, in particular, have the potential to probe the entire viable parameter space of this model.