Accurately detecting and classifying damage in analogue media such as paintings, photographs, textiles, mosaics, and frescoes is essential for cultural heritage preservation. While machine learning models excel in correcting global degradation if the damage operator is known a priori, we show that they fail to predict where the damage is even after supervised training; thus, reliable damage detection remains a challenge. We introduce DamBench, a dataset for damage detection in diverse analogue media, with over 11,000 annotations covering 15 damage types across various subjects and media. We evaluate CNN, Transformer, and text-guided diffusion segmentation models, revealing their limitations in generalising across media types.