Anomalous diffusion phenomena, characterized by non-linear time dependence of mean square displacement, occur extensively in complex systems ranging from biological membranes to financial markets. This study presents a comprehensive analysis of spectral collocation methods for solving fractional differential equations (FDEs) that model anomalous diffusion processes. We develop and implement Chebyshev and Legendre polynomial-based collocation schemes for time-fractional and space-fractional diffusion equations with Caputo and Riemann-Liouville derivatives. Our numerical experiments demonstrate that spectral collocation methods achieve exponential convergence rates for smooth solutions, significantly outperforming traditional finite difference approaches. The proposed algorithms effectively capture the long-range memory effects inherent in fractional operators while maintaining computational efficiency. Stability analysis reveals that the spectral collocation schemes remain stable under appropriate time-stepping constraints. Applications to subdiffusion and superdiffusion scenarios validate the theoretical predictions and demonstrate the practical utility of these methods in modeling real-world anomalous diffusion phenomena.