Compressive sensing is a powerful technique for recovering sparse solutions of underdetermined linear systems, which is often encountered in uncertainty quantification analysis of expensive and high-dimensional physical models. We perform numerical investigations employing several compressive sensing solvers that target the unconstrained LASSO formulation, with a focus on linear systems that arise in the construction of polynomial chaos expansions. With core solvers l1_ls, SpaRSA, CGIST, FPC_AS, and ADMM, we develop techniques to mitigate overfitting through an automated selection of regularization constant based on cross-validation, and a heuristic strategy to guide the stop-sampling decision. Practical recommendations on parameter settings for these techniques are provided and discussed. The overall method is applied to a series of numerical examples of increasing complexity, including large eddy simulations of supersonic turbulent jet-in-crossflow involving a 24-dimensional input. Through empirical phase-transition diagrams and convergence plots, we illustrate sparse recovery performance under structures induced by polynomial chaos, accuracy, and computational trade-offs between polynomial bases of different degrees, and practicability of conducting compressive sensing for a realistic, high-dimensional physical application. Across test cases studied in this paper, we find ADMM to have demonstrated empirical advantages through consistent lower errors and faster computational times.