This paper presents a novel technique for the very efficient and accurate two-dimensional (2-D) spectral analysis of large cylindrical scatterers with arbitrary geometries using the wavelet and wavelet-like transform. When such complex problems are solved through an integral equation (IE) technique combined with the method of moments (MOM), a linear equation system with a huge number of unknowns needs to be solved for each incident field. In this work, wavelet bases are successfully proved to expand the integral operator in a very sparse matrix form, thus providing large savings on computational costs and memory storage requirements. Furthermore, the use of the spectral formulation allows us to characterize the scattering behavior of an object for any possible incidence, thus drastically reducing the number of linear equation systems to be solved. Comparative benchmarks between the new wavelet technique and the original approach for solving the scattering of circular cylinders and strips are presented for different orthogonal families: locally supported Daubechies bases, orthogonal quasi-symmetric Coiflets bases, and wavelet-like bases. Then the efficient wavelet technique is applied to the accurate characterization of more complex geometries, such as large square cylinders.