Shape-memory polymers (SMPs) can adjust their stiffness, lock a temporary shape, and recover the permanent shape upon an appropriate stimulus. They are applied in the field of morphing skins. This work presents a variable-stiffness corrugated sheet based on a carbon fiber felt (CFF)-reinforced epoxy-based SMP composite that shows variable stiffness and extreme mechanical anisotropy for potential morphing skin applications. The corrugated sheet exhibits a variable stiffness with a change in temperature, which can help the skin adjust its stiffness according to different service environments. The corrugated sheet can be electrically heated rapidly and homogeneously due to its high electrical conductivity and enhanced heat transfer efficiency. Its Joule-heating effect acts as an effective active stimulation of the variable stiffness and shape-memory effect. The CFF-reinforced epoxy-based SMP composite was manufactured into a corrugated shape to obtain extreme mechanical anisotropy. The corrugated sheet shows a low in-plane stiffness to minimize the actuation energy, while it also possesses high out-of-plane stiffness to transfer the aerodynamic pressure load. Its mechanical properties, electrical heating performance, and shape-memory effect were investigated using experiments. The results show that the proposed SMP composite exhibits extreme mechanical anisotropy, considerable deformation ability, and variable stiffness induced by Joule heating without an external heater.