The stabilization of two-dimensional non-linear control systems is treated, although the main technique is applicable to more general systems. It combines the theory of centre manifolds and singularities of algebraic functions to design continuous, but not necessarily Lipschitz, stabilizing feedback control laws. In particular, the devices known as the Newton diagram and the Newton or Puiseux polygon are employed to construct uniformizing transformations. Finally, remarks are made about systems for which no continuous stabilizing feedback control law exists.