The tools of Bayesian inference have recently gained interest in tasks of non-rigid registration of medical images, as seminal work demonstrated their potency towards addressing open problems such as the automatic determination of adequate regularization levels or the quantification of confidence in registration outputs. In this paper, we extend the Bayesian modeling of registration to allow for a data-driven, multiscale, spatially adaptive parametrization of deformations. Finer bases get introduced only in the presence of coherent image information and motion, while coarser bases ensure better extrapolation of the motion to textureless, uninformative regions. Adaptive parametrizations have been used with success in the literature to promote both the regularity and accuracy of registration schemes, but so far on non-probabilistic grounds – either as part of multiscale heuristics, or on the basis of sparse optimization. We provide a principled probabilistic approach to find an optimal parametrization of deformations among any preset, widely overcomplete range of basis functions. It thus retains the benefits of the Bayesian formalism, including the estimation of regularization and noise parameters. We further experiment with a richer, more generic model of data that proves to be more faithful for a variety of image modalities than the sum-of-squared differences. We demonstrate the feasibility and performance of our approach on time series of magnetic resonance (cine SSFP and tagged) and echocardiographic cardiac images, and show that the proposed quasi-automatic framework can match or outperform the state-of-the-art on benchmark datasets evaluating accuracy of motion and strain.