We show that van Hove singularities in radio frequency (rf) spectra of the 3D attractive Fermi Hubbard model provide a robust route to extracting the pairing gap. Four types of singularities are classified, and their spectral positions are shown to depend solely on the pairing gap ∆ and chemical potential µ through simple algebraic relations. Measuring two well-resolved singularities therefore determines both parameters without requiring full spectral fitting. Numerical simulations incorporating phenomenological lifetime and scattering broadenings confirm that these features remain visible in both momentum-integrated and kz-integrated spectra, and become more pronounced at stronger coupling where conventional back-bending methods lose sensitivity. At half filling, particle-hole symmetry fixes µ, reducing the extraction to a single singularity measurement. These results establish vHS analysis as a practical spectroscopic diagnostic for pairing in quantum-simulated 3D Fermi Hubbard systems.