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 that 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.

We characterize the extendibility of the normal curvature on frontals and we give a representation formula of this type of frontals. Also we give representation formulas for wavefronts on all types of singularities and others sub classes of these. Some applications to asymptotic curves and lines of curvature on frontals are made.