Microorganisms self-propelling in a fluid create flow fields that impact the dynamics of other swimmers. Some organisms can be biased and have a preferred swimming direction, e.g., those displaying gyrotaxis, chemotaxis or phototaxis, and as a result often focus along thin lines. Here we use numerical computations and far-field theoretical calculations to show that the position of a collection of biased swimmers moving along a line is unstable to a zigzag mode when the swimmers act on the fluid as pusher dipoles. This instability takes the form of periodic transverse oscillations in the position of the swimmers. We predict theoretically that the most unstable wavelength is equal to twice the inter-swimmer distance and that the growth rate of the instability increases linearly with the magnitude of the stresslet, both of which are in quantitative agreement with our numerical simulations.