While numerous efforts have developed and codified AI use case inventories for government in the executive branch, no such inventory exists for the judicial branch. Much focus has been on attorney utilization of AI, but no work has examined the policies courts have instituted to regulate AI use by court employees and judges themselves. We fill this gap in two ways. First, we provide an overview of publicly known AI use cases in the courts, categorizing them by risk and potential impact on judicial discretion. Second, we compile and analyze the internal AI use policies of the 17 U.S. state courts that have promulgated such policies. Our paper proceeds as follows. Section A discusses the methodology for systematically canvassing AI use cases in the courts and state policies. Section B presents an overview of these use cases. Section C analyzes state court policies.