Ship wake detection is of great importance in the characterisation of synthetic aperture radar (SAR) images of the ocean surface since wakes usually carry essential information about vessels. Most detection methods exploit the linear characteristics of the ship wakes and transform the lines in the spatial domain into bright or dark points in a transform domain, such as the Radon or Hough transforms. This paper proposes an innovative ship wake detection method based on sparse regularisation to obtain the Radon transform of the SAR image, in which the linear features are enhanced. The corresponding cost function utilizes the Cauchy prior, and on this basis, the Cauchy proximal operator is proposed. A Bayesian method, the Moreau-Yoshida unadjusted Langevin algorithm (MYULA), which is computationally efficient and robust is used to estimate the image in the transform domain by minimizing the negative log-posterior distribution. The detection accuracy of the Cauchy prior based approach is 86.7%, which is demonstrated by experiments over six COSMO-SkyMed images.