The transient spreading of a viscous fluid beneath an elastic sheet adhered to the substrate is controlled by the dynamics at the tip where the divergence of viscous stresses necessitates the formation of a vapour tip separating the fluid front and fracture front. The model for elastic-plated currents is extended for an axisymmetric geometry with analysis showing that adhesion gives rise to the possibility of static, elastic droplets and to two dynamical regimes of spreading; viscosity dominant spreading controlled by flow of viscous fluid into the vapour tip, and adhesion dominant spreading. Constant flux experiments using clear, PDMS elastic sheets enable new, direct measurements of the vapour tip, and confirm the existence of spreading regimes controlled by viscosity and adhesion. The theory and experiments thereby provide an important test coupling the dynamics of flow with elastic deformation and have implications in fluid-driven fracturing of elastic media more generally.