This study investigates the mechanism of formation of convection plumes of mushroom shape in sub-solidus mantle and their prediction. The seismic-tomographic images of columnar structures of several hundreds kilometers in diameter have been reported by several researchers, while the much cherished mushroom-shaped plume heads could only be found in computational geodynamics (CGD) models and simple small-scale laboratory analogue simulations. Our theory of transient instability shows that the formation of convection plumes is preceded by the onset of convection caused by unsteady-state heat conduction at the boundaries, from which filamentous plumes first appear. The plumes generated at the Core Mantle Boundary (CMB) and lithosphere rising and falling through the mantle have been predicted simply with our theory for various heat fluxes and viscosities, which still remain uncertain amongst geoscientists. The sizes of mushroom plumes in the sub-solidus mantle caused by heat fluxes of 20 and 120 mW/m2 at the CMB are found to be 1842 km and 1173 km with critical times over 825 Myr and 334 Myr respectively. They are comparable to some large continental flood basalt provinces, and they number between 17 and 41. The thickness of the thermal boundary layers at the CMB from which convection plumes evolved are found to be 652 km and 415 km for 20 and 120 mW/m2 respectively. Top cooling may produce plunging plumes of diameter of 585 km and at least 195 Myr old. The number of cold plumes is estimated to be 569, which has not been observed by seismic tomography or as cold spots. The cold plunging plumes may overwhelm and entrap some of the hot rising plumes from CMB, so that together they may settle in the transition zone.