We compare the flow behavior of viscoelastic surfactant (VES) solutions and Newtonian fluids through two different model porous media having similar permeability: (a) a 3D random packed bed and (b) a microchannel with a periodically spaced pillars. The former provides much larger flow resistance at the same apparent shear rate compared to the latter. The flow profile in the 3D packed bed cannot be observed since it is a closed system. However, visualization of the flow profile in the microchannel shows strong spatial and temporal flow instabilities in VES fluids appear above a critical shear rate. The onset of such elastic instabilities correlates to the flow rate where increased flow resistance is observed. The elastic instabilities are attributed to the formation of transient shear induced structures. The experiments provide a detailed insight into the complex interplay between the pore scale geometry and rheology of VES in the creeping flow regime.