Loss of normal tissue architecture is a hallmark of oncogenic transformation. In developing organisms, tissues architectures are sculpted by mechanical forces during morphogenesis. However, the origins and consequences of tissue architecture during tumorigenesis remain elusive. In skin, premalignant basal cell carcinomas form 'buds', while invasive squamous cell carcinomas initiate as 'folds'. Here, using computational modelling, genetic manipulations and biophysical measurements, we identify the biophysical underpinnings and biological consequences of these tumour architectures. Cell proliferation and actomyosin contractility dominate tissue architectures in monolayer, but not multilayer, epithelia. In stratified epidermis, meanwhile, softening and enhanced remodelling of the basement membrane promote tumour budding, while stiffening of the basement membrane promotes folding. Additional key forces stem from the stratification and differentiation of progenitor cells. Tumour-specific suprabasal stiffness gradients are generated as oncogenic lesions progress towards malignancy, which we computationally predict will alter extensile tensions on the tumour basement membrane. The pathophysiologic ramifications of this prediction are profound. Genetically decreasing the stiffness of basement membranes increases membrane tensions in silico and potentiates the progression of invasive squamous cell carcinomas in vivo. Our findings suggest that mechanical forces-exerted from above and below progenitors of multilayered epithelia-function to shape premalignant tumour architectures and influence tumour progression.