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Understanding the mechanisms of shallow magma storage (less than ~10‐km depth) is crucial to defining volcanic plumbing systems and their effect on volcano unrest. Sills are common structures to emplace shallow magma. Many factors may control sill emplacement, but their relative importance has not been evaluated so far. To define a hierarchy among a selection of the proposed factors, we performed analogue models by injecting water at constant flux (magma analogue) within gelatin (crust analogue), investigating the effects of: interface strength and rigidity contrast between layers, density layering, ratio of layer thickness, compressive stresses, magma flow rate, and buoyancy pressure. Our results show that a strong rigidity contrast (i.e., a stiff layer overlying a weak one) and a higher buoyancy pressure are necessary and sufficient conditions for sill emplacement. Low rigidity contrasts require other contributions (weak interface and compressive stress) to develop sills; however, without rigidity contrast sill formation is always inhibited. These experiments suggest that sill emplacement is primarily controlled by rigidity contrasts and magma buoyancy pressure; the second‐order parameters are lateral compression and weak interfaces; the third‐order parameters are the ratio of layer thicknesses, magma flow rate, and density layering. These results are in agreement with field observations in Iceland, Tenerife, Utah, and La Réunion, supporting our proposed hierarchy. Key Points: Understanding what controls a dike to sill rotation is important to improve our knowledge on magma storage in the upper crustWe have defined a hierarchy among selected parameters controlling sill emplacement to understanding shallow magma transferAnalogue models show that high rigidity contrast and magma buoyancy pressure are necessary and sufficient conditions for sill emplacement [ABSTRACT FROM AUTHOR]