By the Fourier inversion numerous operators appearing in mathematical analysis, partial differential equations, and mathematical physics are represented as Fourier multiplier operators. In most cases the associated multipliers have their singularities on certain submanifolds, and regularity properties of such operators are naturally related to the singularities of the multipliers. In this thesis we consider several singular multiplier operators which play important roles in mathematical analysis, and aim to completely characterize mapping properties of these operators on Lebesgue spaces. Our results include sharp estimates for the spherical harmonic projections, sharp resolvent estimates for the Laplacian, and Carleman estimates for second order differential operators. We also investigate applications of the estimates to partial differential equations and spectral theory.