The accuracy of rail profile inspections is critical for guaranteeing transport security and rail maintenance, and hence the laser based rail profile inspection has been frequently used. However, there are two major challenges in practical applications: the distortion of the measured rail profile and the influences of noise and outliers. In this paper, the sparse scaling iterative closest point method is proposed for rail profile inspection. First, the existing challenges for processing the measured rail profile via a line laser sensor are generally described. After this, a robust registration energy function that evolves both the scale factor and lp norm is proposed for rail profile registration. Finally, the Hausdorff distance is adopted to visualize the matching results. The experiments indicate that the proposed method can both precisely rectify the distorted rail profile and avoid the influences of noise and outliers when compared with the conventional iterative closest point, sparse iterative closest point and reweighted-scaling closest point methods.