Sparse polynomial chaos expansions (PCE) are an efficient and widely used surrogate modeling method in uncertainty quantification for engineering problems with computationally expensive models. To make use of the available information in the most efficient way, several approaches for so-called basis-adaptive sparse PCE have been proposed to determine the set of polynomial regressors ("basis") for PCE adaptively. We describe three state-of-the-art basis-adaptive approaches from the recent sparse PCE literature and extensively benchmark them in terms of global approximation accuracy on a large set of computational models representative of a wide range of engineering problems. Investigating the synergies between sparse regression solvers and basis adaptivity schemes, we find that virtually all basis-adaptive schemes outperform a static choice of basis. Three sparse solvers, namely Bayesian compressive sensing and two variants of subspace pursuit, perform especially well. Aggregating our results by model dimensionality and experimental design size, we identify combinations of methods that are most promising for the specific problem class. Additionally, we introduce a novel solver and basis adaptivity selection scheme guided by cross-validation error. We demonstrate that this meta-selection procedure provides close-to-optimal results in terms of accuracy, and significantly more robust solutions, while being more general than the case-by-case recommendations obtained by the benchmark.