Deep-learning methods have recently started being employed for addressing surface-defect detection problems in industrial quality control. However, with a large amount of data needed for learning, often requiring high-precision labels, many industrial problems cannot be easily solved, or the cost of the solutions would significantly increase due to the annotation requirements. In this work, we relax heavy requirements of fully supervised learning methods and reduce the need for highly detailed annotations. By proposing a deep-learning architecture, we explore the use of annotations of different details ranging from weak (image-level) labels through mixed supervision to full (pixel-level) annotations on the task of surface-defect detection. The proposed end-to-end architecture is composed of two sub-networks yielding defect segmentation and classification results. The proposed method is evaluated on several datasets for industrial quality inspection: KolektorSDD, DAGM and Severstal Steel Defect. We also present a new dataset termed KolektorSDD2 with over 3000 images containing several types of defects, obtained while addressing a real-world industrial problem. We demonstrate state-of-the-art results on all four datasets. The proposed method outperforms all related approaches in fully supervised settings and also outperforms weakly-supervised methods when only image-level labels are available. We also show that mixed supervision with only a handful of fully annotated samples added to weakly labelled training images can result in performance comparable to the fully supervised model's performance but at a significantly lower annotation cost.