This paper is concerned with control systems on the plane with control appearing linearly. It is known that under generic conditions the problem of reaching points from the origin in minimum time admits a regular synthesis. The minimum time function is piecewise smooth, possibly nondifferentiable on a set that is a finite union of embedded submanifolds of dimension 1 or 0, called singularities. The purpose of the present paper is to provide a classification of all types of singularities arising under generic conditions.