This paper investigates the singularities of five special surfaces generated by a regular curve lying on a spacelike hypersurface in Lorentz–Minkowski 4-space. The authors use extended Lorentzian Darboux frames along the curve to derive five new invariants that characterize the singularities of these surfaces. These invariants are detailed and their geometric meanings are discussed. Additionally, the paper reveals dual relationships between the normal curve of the original curve and the five surfaces under the concept of Legendrian duality. The study fills a gap in the literature by analyzing the singularities of surfaces generated by curves on spacelike hypersurfaces, which have not been previously explored. The main results include the classification of the singularities into cuspidal edge and swallowtail types, and the presentation of the Legendrian dualities between the surfaces and the curve. The paper also provides proofs using unfolding theory and discusses the geometric interpretations of the invariants.This paper investigates the singularities of five special surfaces generated by a regular curve lying on a spacelike hypersurface in Lorentz–Minkowski 4-space. The authors use extended Lorentzian Darboux frames along the curve to derive five new invariants that characterize the singularities of these surfaces. These invariants are detailed and their geometric meanings are discussed. Additionally, the paper reveals dual relationships between the normal curve of the original curve and the five surfaces under the concept of Legendrian duality. The study fills a gap in the literature by analyzing the singularities of surfaces generated by curves on spacelike hypersurfaces, which have not been previously explored. The main results include the classification of the singularities into cuspidal edge and swallowtail types, and the presentation of the Legendrian dualities between the surfaces and the curve. The paper also provides proofs using unfolding theory and discusses the geometric interpretations of the invariants.