Likelihood-based tests of phylogenetic trees are a foundation of modern systematics, and examples of such tests are among the most widely-referenced scientific literature of all time. Over the past decade, an enormous wealth and diversity of model-based approaches have been developed for phylogenetic inference of both gene trees and species trees. However, while many techniques exist for conducting formal likelihood-based tests of gene trees, such frameworks are comparatively underdeveloped and underutilized for testing species-level hypotheses. To date, widely-used tests of tree topology are designed to assess the fit of classical models of molecular sequence data and individual gene trees, and thus, are not readily applicable to the problem of species tree inference. We derive several analogous likelihood-based approaches for testing species-level topologies using modern species tree models and algorithms for maximum likelihood estimation under the multispecies coalescent. For the purpose of comparing support for species-level relationships, these tests leverage the statistical procedures of their original gene tree-based counterparts that have a long history for testing phylogenetic hypotheses at single loci. We discuss and demonstrate a number of applications, limitations, and important considerations of these tests using simulated and empirical phylogenomic datasets that include both bifurcating topologies and reticulate network models of species relationships. Finally, we introduce the open-source R package SpeciesTopoTestR (Species Topology Tests in R) that includes a suite of functions for conducting formal likelihood-based tests of species topologies.