Interpreting and predicting basin-margin stratal geometries requires understanding of controls such as variations in supply and accommodation, ideally based on independent quantitative evidence. Stratal-control spaces are a new tool to analyze controls on strata. A stratal-control space is an area, volume, or perhaps a higher-dimensional space, defined by a range of values of the controlling processes subsidence, sediment supply, and eustasy. A three-dimensional stratal-control volume with axes of subsidence, sediment supply, and eustatic rates of change can be populated with probabilities derived from analysis of time series of subsidence, supply, and eustasy. These empirical or theoretical probabilities indicate the likelihood of occurrence of any particular combination of control rates defined by any point in the volume. The stratal-control volume can then be analyzed to determine which parts of the volume represent relative sea-level fall and rise, where in the volume particular stacking patterns will occur, and how probable those stacking patterns are. For outcrop and subsurface analysis, using a stratal-control area with eustasy and subsidence combined on a relative sea-level axis allows similar analysis, and may be preferable. A stratal-control trajectory is a history of supply and accommodation rates, interpreted from outcrop or subsurface data, or observed in analogue and numerical experiments, and plotted as a series of linked points forming a trajectory through a stratal-control space. Two theoretical and one actual outcrop example are presented to demonstrate how stratal-control trajectories can be analyzed to determine which controls are dominant. The accommodation supply trajectory range ratio (ASTRR) is a useful metric to characterize trajectory geometry. Trajectories with ASTRR > 1 can be considered accommodation-dominated, and ASTRR < 1 indicates a supply-dominated trajectory. Calculating the range of stratal-control probabilities along the trajectory indicates the probability of the rates of change of subsidence, supply, and eustasy required to form the interpreted stratal geometry. Both types of stratal-control-trajectory analyses can provide important additional understanding and prediction of how, why, and where stratal geometries form.