Correctly resolving textual mentions of people fundamentally entails making inferences about those people. Such inferences raise the risk of systematic biases in coreference resolution systems, including biases that can harm binary and non-binary trans and cis stakeholders. To better understand such biases, we foreground nuanced conceptualizations of gender from sociology and sociolinguistics, and investigate where in the machine learning pipeline such biases can enter a coreference resolution system. We inspect many existing data sets for trans-exclusionary biases, and develop two new data sets for interrogating bias in both crowd annotations and in existing coreference resolution systems. Through these studies, conducted on English text, we confirm that without acknowledging and building systems that recognize the complexity of gender, we will build systems that fail for: quality of service, stereotyping, and over- or under-representation, especially for binary and non-binary trans users.