Texture segmentation constitutes a standard image processing task, crucial for many applications. The present contribution focuses on the particular subset of scale-free textures and its originality resides in the combination of three key ingredients: First, texture characterization relies on the concept of local regularity; Second, estimation of local regularity is based on new multiscale quantities referred to as wavelet leaders; Third, segmentation from local regularity faces a fundamental bias variance tradeoff. In nature, local regularity estimation shows high variability that impairs the detection of changes, while a posteriori smoothing of regularity estimates precludes from locating correctly changes. Instead, the present contribution proposes several variational problem formulations based on total variation and proximal resolutions that effectively circumvent this tradeoff. Estimation and segmentation performance for the proposed procedures are quantified and compared on synthetic as well as on real-world textures.