Surface instabilities in a capillary extrusion have been studied for various ethylene/alpha-olefin copolymers. It is found that the onset stress of shark-skin failure for ethylene/1-hexene copolymer (EHR) decreases rapidly with increasing 1-hexene content, whereas that of ethylene/propylene copolymer (EPR) is independent of propylene content in the experimental region. Consequently, EHR with high 1-hexene content exhibits shark-skin at low stress level compared to EPR. Lower rubbery plateau modulus, leading to higher Deborah number at the same stress level, is attributed to the lower onset stress. Further, the low entanglement density will cause cracks at lower stress level like glassy polymers, which is also responsible for the low onset stress for shark-skin.