When approximating PDE eigenvalue problems by numerical methods such as finite difference and finite element, it is common knowledge that only a small portion of numerical eigenvalues are reliable. As a comparison, spectral methods may perform extremely well in some situation, especially for 1-D problems and some special higher dimensional situations. In addition, we demonstrate that spectral methods can outperform traditional methods and the state-of-the-art method in 2-D problems even with singularities.