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We perform a systematic study of the $d=5$ Weinberg operator at the one-looplevel. We identify three different categories of neutrino mass generation: (1)finite irreducible diagrams; (2) finite extensions of the usual seesawmechanisms at one-loop and (3) divergent loop realizations of the seesaws. Allradiative one-loop neutrino mass models must fall into one of these classes.Case (1) gives the leading contribution to neutrino mass naturally and aclassic example of this class is the Zee model. We demonstrate that in order toprevent that a tree level contribution dominates in case (2), Majorana fermionsrunning in the loop and an additional $\mathbb{Z}_2$ symmetry are needed for agenuinely leading one-loop contribution. In the type-II loop extensions, thelepton number violating coupling will be generated at one loop, whereas thetype-I/III extensions can be interpreted as loop-induced inverse or linearseesaw mechanisms. For the divergent diagrams in category (3), the tree levelcontribution cannot be avoided and is in fact needed as counter term to absorbthe divergence.

Systematic study of the d=5 Weinberg operator at one-loop order