Metropolis algorithms are classical tools for sampling from target distributions, with broad applications in statistics and scientific computing. Their convergence speed is governed by the spectral gap of the associated Markov operator. Recently, Andrieu et al. (2024) derived the first explicit bounds for the spectral gap of Random--Walk Metropolis when the target distribution is smooth and strongly log-concave. However, existing literature rarely discusses non-smooth targets. In this work, we derive explicit spectral gap bounds for the random-walk Metropolis and Metropolis--adjusted Langevin algorithms over a broad class of non-smooth distributions. Moreover, combining our analysis with a recent result in Goyal et al. (2025), we extend these bounds to targets satisfying a Poincare or log-Sobolev inequality, beyond the strongly log-concave setting. Our theoretical results are further supported by numerical experiments.