The structure and deformation characteristics of ethylene-vinyl acetate (EVA)-layered silicate nanocomposites were studied using steady shear and extensional rheology. EVAs with VA contents of 18% and 28% by weight and a commercially modified montmorillonite clay were melt compounded in a twin-screw extruder. Nanocomposites of 2.5%, 5% and 7.5% by weight were produced. WAXS measurements have revealed that EVA chains had intercalated into the silicate layers and expanded the interlayer distance. TEM showed that the morphologies of the nanocomposites were of mixed intercalated/exfoliated. The nanocomposites exhibited an increase in steady shear viscosities compared to the unfilled EVA polymers. At high loading there was a possibility of yield stress owing to network structures. Elasticity of the nanocomposites was compared using first normal stress difference-shear stress plots and the modified Cole-Cole plots, with both methods showing contradictory results. Melt strengths of both EVAs were enhanced with the addition of silicate fillers; however, this was at the expense of the materials' extensibility. Addition of silicate fillers also brought with it flow instabilities like draw resonance and ductile failure, both of which lead to limitations in polymer processing.