The rapid proliferation of trajectory inference methods for single-cell RNA-seq data has allowed researchers to investigate complex biological processes by examining underlying gene expression dynamics. After estimating a latent cell ordering, statistical models are used to determine which genes exhibit changes in expression that are significantly associated with progression through the biological trajectory. While a few techniques for performing trajectory differential expression exist, most rely on the flexibility of generalized additive models in order to account for the inherent nonlinearity of changes in gene expression. As such, the results can be difficult to interpret, and biological conclusions often rest on subjective visual inspections of the most dynamic genes. To address this challenge, we propose scLANE testing, which is built around an interpretable generalized linear model and handles nonlinearity with basis splines chosen empirically for each gene. In addition, extensions to estimating equations and mixed models allow for reliable trajectory testing under complex experimental designs. After validating the accuracy of scLANE under several different simulation scenarios, we apply it to a set of diverse biological datasets and display its ability to provide novel biological information when used downstream of both pseudotime and RNA velocity estimation methods.