We used seven scaled physical models to explore the near-surface structural evolution of shallowly buried, actively rising salt stocks. The models consisted of dry sand, ceramic microspheres and silicone. Previously dormant stocks rose because of lateral squeezing or pumping of salt from below. The pressure of rising salt created a dynamic bulge in the crest of the diapir, which arched the overlying roof sediments. Eventually this dynamic bulge collapsed and its overlying roof broke into rafts along subradial grabens. The rafts were dispersed outwards by shear traction of spreading salt, surmounting an upturned collar of country rock and eventually grounding at the front of the extrusive flow. Flow of salt around these stranded fragments created a lobate extrusion front, common in submarine salt sheets in the Gulf of Mexico and subaerial salt glaciers in Iran. Stock geometry, regional dip and roof density affected extrusion rates and spreading directions. Stocks leaning seaward extruded salt faster and farther than did upright stocks. Dense roofs foundered and plugged the vent, limiting surface extrusion. In tilted models, broad salt sheets spread asymmetrically downslope. Stock contents were inverted within the extruded salt sheet: successively deeper parts of the stock's core rose to the surface and overran salt extruded from the shallower parts of the diapir. As shortening continued, salt from the source layer reached the surface after being driven out by thrusting. A central thrust block, or primary indenter, moved ahead of surrounding thrust blocks, impinging against and squeezing the stock into an elliptical planform. After high shortening, secondary indenters converged obliquely into the salt stock, expelling salt from the periphery of the diapir. The models shed light on (1) the origin and fate of large rafts or carapace blocks atop allochthonous salt, (2) cuspate margins of salt sheets and (3) interaction of thrusting, diapir pinch-off and emplacement of allochthonous salt sheets.