The aim of this paper is to study several strategies for the preservation of flow discontinuities in variational optical flow methods. We analyze the combination of robust functionals and diffusion tensors in the smoothness assumption. Our study includes the use of tensors based on decreasing functions, which has shown to provide good results. However, it presents several limitations and usually does not perform better than other basic approaches. It typically introduces instabilities in the computed motion fields in the form of independent blobs of vectors with large magnitude. We propose two alternatives to overcome these drawbacks: first, a simple approach that combines the decreasing function with a minimum isotropic smoothing, and second, a method that looks for the best parameter configuration that preserves the important motion contours and avoid instabilities. It relies on the input images and the regularization parameter. It is fully automatic, providing a near-optimal value for many sequences, as shown in the experiments. Both proposals allow to detect the contours of the motion field and produce more stable solutions for a large range of parameters. In the experimental results, we present a detailed study and comparison of the different strategies.