In the context of scene understanding, a variety of methods exists to estimate different information channels from mono or stereo images, including disparity, depth, and normals. Although several advances have been reported in the recent years for these tasks, the estimated information is often imprecise particularly near depth discontinuities or creases. Studies have however shown that precisely such depth edges carry critical cues for the perception of shape, and play important roles in tasks like depth-based segmentation or foreground selection. Unfortunately, the currently extracted channels often carry conflicting signals, making it difficult for subsequent applications to effectively use them. In this paper, we focus on the problem of obtaining high-precision depth edges (i.e., depth contours and creases) by jointly analyzing such unreliable information channels. We propose DepthCut, a data-driven fusion of the channels using a convolutional neural network trained on a large dataset with known depth. The resulting depth edges can be used for segmentation, decomposing a scene into depth layers with relatively flat depth, or improving the accuracy of the depth estimate near depth edges by constraining its gradients to agree with these edges. Quantitatively, we compare against 18 variants of baselines and demonstrate that our depth edges result in an improved segmentation performance and an improved depth estimate near depth edges compared to data-agnostic channel fusion. Qualitatively, we demonstrate that the depth edges result in superior segmentation and depth orderings. (Code and datasets will be made available.)