In this paper, the growth rate and eigenmode structures of the streaming sausage and kink instabilities in a current sheet with a super‐Alfvénic flow are studied. Based on the linearized compressible MHD equations, we employ an initial‐value method to obtain the growth rate and eigenmode profiles of the fastest growing mode which is either the streaming sausage mode or kink mode. The streaming sausage and kink instabilities, similar to the Kelvin‐Helmholtz instability, are caused by the sheared plasma flow. The results show that for V0m = 2 VA∞ the sausage mode grows faster than the kink mode when β∞ < 1.5. When β∞ > 1.5, the streaming kink instability has a higher growth rate. Here VA∞ is the Alfvén velocity and β∞ is the ratio between the plasma and magnetic pressures far away from the current layer, and V0m is the maximum velocity of plasma flow at the current sheet. In addition, an analytical dispersion equation is obtained for an ideal four‐layer model of the current sheet in the incompressible limit. In the presence of a finite resistivity, the mixed sausage‐tearing mode or the streaming tearing mode may be excited, which leads to the formation of plasmoids in the magnetotail. As an application to the Earth's magnetotail, where super‐Alfvénic plasma flows are observed in the plasma sheet and β∞ ≃ 0.1–0.3 in the lobes, it is suggested that the streaming sausage and streaming tearing instabilities may occur in the magnetotail. Some of the north‐then‐south or south‐then‐north Bz signatures observed in the distant magnetotail may be associated with the streaming sausage mode or the mixed sausage‐tearing mode. It is also pointed out that kink modes are unlikely to occur in the magnetotail.