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Abstract The interaction of supernova ejecta with a surrounding circumstellar medium (CSM) generates a strong shock, which can convert ejecta kinetic energy into observable radiation. Given the diversity of potential CSM structures (arising from diverse mass-loss processes such as late-stage stellar outbursts, binary interaction, and winds), the resulting transients can display a wide range of light-curve morphologies. We provide a framework for classifying the transients arising from interaction with a spherical CSM shell. The light curves are decomposed into five consecutive phases, starting from the onset of interaction and extending through shock breakout and subsequent shock cooling. The relative prominence of each phase in the light curve is determined by two dimensionless quantities representing the CSM-to-ejecta mass ratioη, and the breakout parameterξ. These two parameters define four light-curve morphology classes, where each class is characterized by the location of the shock breakout and the degree of deceleration as the shock sweeps up the CSM. We compile analytic scaling relations connecting the luminosity and duration of each light-curve phase to the physical parameters. We then run a grid of radiation hydrodynamics simulations for a wide range of ejecta and CSM parameters to numerically explore the landscape of interaction light curves, and to calibrate and confirm the analytic scalings. We connect our theoretical framework to several case studies of observed transients, highlighting the relevance in explaining slow-rising and superluminous supernovae, fast blue optical transients, and double-peaked light curves.