This paper investigates a discretization scheme for mean curvature motion on point cloud varifolds with particular emphasis on singular evolutions. To define the varifold a local covariance analysis is applied to compute an approximate tangent plane for the points in the cloud. The core ingredient of the mean curvature motion model is the regularization of the first variation of the varifold via convolution with kernels with small stencil. Consistency with the evolution velocity for a smooth surface is proven if a sufficiently small stencil and a regular sampling are taking into account. Furthermore, an implicit and a semiimplicit time discretization are derived. The implicit scheme comes with discrete barrier properties known for the smooth, continuous evolution, whereas the semiimplicit still ensures in all our numerical experiments very good approximation properties while being easy to implement. It is shown that the proposed method is robust with respect to noise and recovers the evolution of smooth curves as well as the formation of singularities such as triple points in 2D or minimal cones in 3D.