If the axiom of foundation is not assumed (that is, is not in S) all three of these concepts are given the additional condition that N be well-founded.
The minimal submodel contains no standard submodel (as it is minimal) but (assuming the consistency of ZFC) it contains some model of ZFC by the Gödel completeness theorem.
This model is necessarily not well-founded otherwise its Mostowski collapse would be a standard submodel.
However, it is not uncommon to talk about inner models of subtheories of ZFC (like ZF or KP) as well.
Inner model theory has led to the discovery of the exact consistency strength of many important set theoretical properties.