This paper investigates the singularities of five special surfaces generated by a regular curve lying on a spacelike hypersurface in Lorentz–Minkowski 4-space. Using two kinds of extended Lorentzian Darboux frames along the curve, five new invariants are obtained to characterize the singularities of these surfaces and their geometric meanings are discussed. The study also reveals some dual relationships between a normal curve of the original curve and the five surfaces under the meanings of Legendrian duality. Curves and surfaces are common research objects in differential geometry and low-dimensional topology. Frame fields associated with a space curve serve as a useful tool in the study of curves and surfaces. Duldul et al. generalize the Darboux frame into 4-space along a Frenet curve lying on an oriented hypersurface and call this new frame field an extended Darboux frame field. In recent years, singularity theory has developed rapidly in both theory and application. Izumiya et al. have studied the singularities of smooth mappings in semi-Euclidean space, especially the singularities of surfaces generated by original curves lying on non-constant curvature space forms. They generalized the Darboux frame in Euclidean 3-space into Minkowski 3-space and established the frame field called Lorentzian Darboux frame along the original curves. However, no literature exists regarding the singularities of surfaces as they relate to spacelike curves in non-constant curvature submanifolds of dimension at least 3. The current study focuses on analyzing the singularities of surfaces generated by original curves lying on spacelike hypersurfaces. Corresponding to the extended Darboux frame in Euclidean 4-space, we establish a Lorentzian version's frame called an extended Lorentzian Darboux frame. Several new invariants and five height functions are defined in this paper. Applying unfolding theory in singularity theory to five height functions, the singularities of five surfaces are classified into two types: cuspidal edge and swallowtail. These singularities are also characterized by several invariants. The paper reviews the basic notions of Lorentz–Minkowski space, defines five surfaces along curves lying in a spacelike surface, and establishes the extended Darboux frame along the curves. The geometric meanings of invariants are discussed, and some examples when curves lie on constant curvature space forms are provided to illustrate the theoretical results.This paper investigates the singularities of five special surfaces generated by a regular curve lying on a spacelike hypersurface in Lorentz–Minkowski 4-space. Using two kinds of extended Lorentzian Darboux frames along the curve, five new invariants are obtained to characterize the singularities of these surfaces and their geometric meanings are discussed. The study also reveals some dual relationships between a normal curve of the original curve and the five surfaces under the meanings of Legendrian duality. Curves and surfaces are common research objects in differential geometry and low-dimensional topology. Frame fields associated with a space curve serve as a useful tool in the study of curves and surfaces. Duldul et al. generalize the Darboux frame into 4-space along a Frenet curve lying on an oriented hypersurface and call this new frame field an extended Darboux frame field. In recent years, singularity theory has developed rapidly in both theory and application. Izumiya et al. have studied the singularities of smooth mappings in semi-Euclidean space, especially the singularities of surfaces generated by original curves lying on non-constant curvature space forms. They generalized the Darboux frame in Euclidean 3-space into Minkowski 3-space and established the frame field called Lorentzian Darboux frame along the original curves. However, no literature exists regarding the singularities of surfaces as they relate to spacelike curves in non-constant curvature submanifolds of dimension at least 3. The current study focuses on analyzing the singularities of surfaces generated by original curves lying on spacelike hypersurfaces. Corresponding to the extended Darboux frame in Euclidean 4-space, we establish a Lorentzian version's frame called an extended Lorentzian Darboux frame. Several new invariants and five height functions are defined in this paper. Applying unfolding theory in singularity theory to five height functions, the singularities of five surfaces are classified into two types: cuspidal edge and swallowtail. These singularities are also characterized by several invariants. The paper reviews the basic notions of Lorentz–Minkowski space, defines five surfaces along curves lying in a spacelike surface, and establishes the extended Darboux frame along the curves. The geometric meanings of invariants are discussed, and some examples when curves lie on constant curvature space forms are provided to illustrate the theoretical results.