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New technologies and the resulting changes to regulatory approaches have led to an explosion in the complexity and crowding of radio spectrum. This rapidly evolving landscape has led to the increased importance of two fields: interference management and dynamic spectrum sharing. However, the progress of both areas is hampered by the lack of shared vocabularies, which, if developed, could help to streamline communication between relevant parties in government, industry, and academia. This paper first presents a taxonomy to aid in the description of key spectrum sharing concepts and architectures to achieve a similar goal. Next, it defines a hierarchical taxonomy of physical-layer interference as an educational and practical tool for detection, identification, location, reporting, mitigation, and remediation of interference. Finally, the use of these two taxonomies is demonstrated through the application of the classification systems to spectrum sharing regimes and prominent case studies.