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Most image segmentation algorithms are trained on binary masks formulated as a classification task per pixel. However, in applications such as medical imaging, this "black-and-white" approach is too constraining because the contrast between two tissues is often ill-defined, i.e., the voxels located on objects' edges contain a mixture of tissues. Consequently, assigning a single "hard" label can result in a detrimental approximation. Instead, a soft prediction containing non-binary values would overcome that limitation. We introduce SoftSeg, a deep learning training approach that takes advantage of soft ground truth labels, and is not bound to binary predictions. SoftSeg aims at solving a regression instead of a classification problem. This is achieved by using (i) no binarization after preprocessing and data augmentation, (ii) a normalized ReLU final activation layer (instead of sigmoid), and (iii) a regression loss function (instead of the traditional Dice loss). We assess the impact of these three features on three open-source MRI segmentation datasets from the spinal cord gray matter, the multiple sclerosis brain lesion, and the multimodal brain tumor segmentation challenges. Across multiple cross-validation iterations, SoftSeg outperformed the conventional approach, leading to an increase in Dice score of 2.0% on the gray matter dataset (p=0.001), 3.3% for the MS lesions, and 6.5% for the brain tumors. SoftSeg produces consistent soft predictions at tissues' interfaces and shows an increased sensitivity for small objects. The richness of soft labels could represent the inter-expert variability, the partial volume effect, and complement the model uncertainty estimation. The developed training pipeline can easily be incorporated into most of the existing deep learning architectures. It is already implemented in the freely-available deep learning toolbox ivadomed (https://ivadomed.org).