  In this paper, we prove a similar result to the fundamental theorem of
regular surfaces in classical differential geometry, which extends the
classical theorem to the entire class of singular surfaces in Euclidean 3-space
known as frontals. Also, we characterize in a simple way these singular
surfaces and its fundamental forms with local properties in the differential of
its parametrization and decompositions in the matrices associated to the
fundamental forms. In particular we introduce new types of curvatures which can
be used to characterize wave fronts. The only restriction on the
parametrizations that is assumed in several occasions is that the singular set
has empty interior.
