There are generally the mixed type surfaces with lightlike locus in the Lorentz-Minkowski 3-space. To investigate the geometry of lightlike locus, we define slant focal surfaces and slant evolutes associated to the oringinal mixed type surface by using a moving frame field along the lightlike locus defined by Honda etc. We obtain that singularities of slant focal surfaces and slant evolutes depend on the differential geometric properties of the lightlike locus. Furthermore, we investigate the relationship between slant focal surfaces and slant evolutes. We also consider the relationship between slant evolutes and the lightlike locus on the lightcone.

Cuspidal edges represent the generic singularities of wave fronts. We apply Legendrian dualities, originally introduced by Izumiya in the Minkowski space and subsequently extended to the semi-Euclidean space by Izumiya and the third author of the paper, to establish the relations of principal curvatures between cuspidal edges and their dual surfaces.Additionally, we determine the relationships among several parallel surfaces associated with cuspidal edges and their corresponding dual surfaces.Based on the aforementioned two types of relations, we explore the connections between the singularity types of parallel surfaces and the geometric characteristics of the initial cuspidal edges, as well as those of the dual surfaces of the initial cuspidal edges, respectively. Finally, we analyze the correspondence between the singularities and geometric properties of parallel surfaces that are Legendrian dual to each other.