In a weighted modal logics framework, this paper studies the definition of weighted extensions for the classical modal axioms. It discusses the notion of relevant weight values, in a specific weighted Kripke semantics and exploiting accessibility relation properties. Different generalisations of the classical axioms are constructed and, from these, a typology of weighted axioms is built, distinguishing between four types, depending on their relations to their classical counterparts and to the, possibly equivalent, frame conditions.