We introduce a novel scale-space concept that is inspired by inpainting-based lossy image compression and the recent denoising by inpainting method of Adam et al. (2017). In the discrete setting, the main idea behind these so-called sparsification scale-spaces is as follows: Starting with the original image, one subsequently removes a pixel until a single pixel is left. In each removal step the missing data are interpolated with an inpainting method based on a partial differential equation. We demonstrate that under fairly mild assumptions on the inpainting operator this general concept indeed satisfies crucial scale-space properties such as gradual image simplification, a discrete semigroup property or invariances. Moreover, our experiments show that it can be tailored towards specific needs by selecting the inpainting operator and the pixel sparsification strategy in an appropriate way. This may lead either to uncommitted scale-spaces or to highly committed, image-adapted ones.