Discrete vine copulas provide a flexible modeling framework for high-dimensional data and have significant computational advantages over competing methods. A vine-based multivariate probability mass function is constructed from bivariate copula building blocks and univariate marginal distributions. However, even for a moderate number of variables, the number of alternative vine decompositions is very large and additionally there is a large set of candidate bivariate copula families that can be used as building blocks in any given decomposition. Together, these two issues ensure that it is infeasible to evaluate all possible vine copula models. Instead, two greedy algorithms for automatically selecting vine structures and component pair-copula building blocks are introduced. The algorithms are tested in a simulation study that is itself driven by real world data from online retail. Both algorithms select vines that provide accurate estimates of the joint probabilities. Using three different f-divergences as criteria, the proposed algorithms outperform a Gaussian copula benchmark, especially for data with high dependence. Finally, the selection algorithm is applied to data from the General Social Survey and outperforms a Gaussian copula benchmark using both in-sample and out-of-sample criteria.