In the presence of a vertically varying horizontal current (background shear), the salt-fingering instability is supplanted by the salt-sheet instability. Previous direct numerical simulation (DNS) experiments on salt sheets revealed that flow becomes turbulent via secondary instabilities. We call these instabilities zig-zag and tip modes. Here, we investigate the physics of these modes using linear normal mode stability analysis. As the primary instability (salt-sheet instability) grows, the zig-zag mode emerges, which denotes undulation of growing salt sheets at the center of fingering regions. This mode is shown to be an extension of secondary instability of unsheared two-dimensional salt-fingering. The zig-zag mode is amplifed almost uniformly at all horizontal wavelengths exceeding O(1 m). This mechanism may, therefore, account for the tilted laminae seen in shadowgraph images of microstructure in salt fingering regions. Subsequently, the tip mode appears at the tips of undulating salt sheets introducing streamwise dependence that leads the flow into turbulent regime.