We are now in the era of millimeter-scale electron microscopy (EM) volumes collected at nanometer resolution (Shapson-Coe et al., 2021; Consortium et al., 2021). Dense reconstruction of cellular compartments in these EM volumes has been enabled by recent advances in Machine Learning (ML) (Lee et al., 2017; Wu et al., 2021; Lu et al., 2021; Macrina et al., 2021). Automated segmentation methods can now yield exceptionally accurate reconstructions of cells, but despite this accuracy, laborious post-hoc proofreading is still required to generate large connectomes free of merge and split errors. The elaborate 3-D meshes of neurons produced by these segmentations contain detailed morphological information, from the diameter, shape, and branching patterns of axons and dendrites, down to the fine-scale structure of dendritic spines. However, extracting information about these features can require substantial effort to piece together existing tools into custom workflows. Building on existing open-source software for mesh manipulation, here we present "NEURD", a software package that decomposes each meshed neuron into a compact and extensively-annotated graph representation. With these feature-rich graphs, we implement workflows for state of the art automated post-hoc proofreading of merge errors, cell classification, spine detection, axon-dendritic proximities, and other features that can enable many downstream analyses of neural morphology and connectivity. NEURD can make these new massive and complex datasets more accessible to neuroscience researchers focused on a variety of scientific questions.