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We study the conceptual details and the physical interpretation of the two prominent singularity theorems due to Penrose and Hawking. Their usage is discussed in detail for the Schwarzschild spacetime with positive and negative mass. First, we present a detailed mathematical proof to formally guarantee the existence of a singularity of geodesic incompleteness for the case of positive mass. Second, we discuss the applicability of the mathematical tools used by the theorems in the negative mass case. The physical implications of the validity or inconsistency of the hypotheses of such theorems on the latter case are also exhibited. As far as this analysis is concerned, some clues are produced regarding future research that could result in general properties for naked singularities.