Inverse problems provide a fundamental framework for image reconstruction tasks, spanning deblurring, calibration, or low-light enhancement for instance. While widely used, they often assume full knowledge of the forward model---an unrealistic expectation---while collecting ground truth and measurement pairs is time-consuming and labor-intensive.Without paired supervision or an invertible forward model, solving inverse problems becomes significantly more challenging and error-prone. To address this, strong priors have traditionally been introduced to regularize the problem, enabling solutions from single images alone.In this work, however, we demonstrate that with minimal assumptions on the forward model and by leveraging small, unpaired clean and degraded datasets, we can achieve good estimates of the true degradation. We employ conditional flow matching to efficiently model the degraded data distribution and explicitly learn the forward model using a tailored distribution-matching loss.Through experiments on uniform and non-uniform deblurring tasks, we show that our method outperforms both single-image blind and unsupervised approaches, narrowing the gap to non-blind methods. We also showcase the effectiveness of our method with a proof of concept for automatic lens calibration---a real-world application traditionally requiring time-consuming experiments and specialized equipment. In contrast, our approach achieves this with minimal data acquisition effort.