In this paper, we present three deformable registration algorithms designed within a paradigm that uses 3D statistical shape models to accomplish two tasks simultaneously: 1) register point features from previously unseen data to a statistically derived shape (e.g., mean shape), and 2) deform the statistically derived shape to estimate the shape represented by the point features. This paradigm, called the deformable most-likely-point paradigm, is motivated by the idea that generative shape models built from available data can be used to estimate previously unseen data. We developed three deformable registration algorithms within this paradigm using statistical shape models built from reliably segmented objects with correspondences. Results from several experiments show that our algorithms produce accurate registrations and reconstructions in a variety of applications with errors up to CT resolution on medical datasets. Our code is available at https://github.com/AyushiSinha/cisstICP.