The notion of framed slant helices is presented in a way which allows the existence of singular points. Some characterizations of the framed slant helices are described via certain equivalent conditions. The framed principal-directional curve is introduced as a framed curve and an associated curve of the original curve. It is revealed that there exists an extra relationship between the framed principal-directional curve and the original curve when the framed principal-donor curve is considered as a framed slant helix. Meanwhile, we consider two surfaces, the principal normal rectifying developable surface and the Darboux normal developable surface, generated by a Frenet type framed base curve, adopting the method of unfolding theory and using two new geometric invariants, which reflect the information about the types of singularities of these two surfaces, we distinguish the generic cuspidal edge and swallowtail type of singularities for such two surfaces. In particular, the principal normal rectifying developable surface will be a cylinder surface if the original curve is a framed slant helix. Finally, several examples are provided to illustrate the main results.