We investigate in this work a versatile convex framework for multiple image segmentation, relying on the regularized optimal mass transport theory. In this setting, several transport cost functions are considered and used to match statistical distributions of features. In practice, global multidimensional histograms are estimated from the segmented image regions and are compared to reference models that are either fixed histograms given a priori, or directly inferred in the non-supervised case. The different convex problems studied are solved efficiently using primal--dual algorithms. The proposed approach is generic and enables multiphase segmentation as well as co-segmentation of multiple images.