Registration uncertainty may be important information to convey to a surgeon when surgical decisions are taken based on registered image data. However, conventional non-rigid registration methods only provide the most likely deformation. In this paper we show how to determine the registration uncertainty, as well as the most likely deformation, by using an elastic Bayesian registration framework that generates a dense posterior distribution on deformations. We model both the likelihood and the elastic prior on deformations with Boltzmann distributions and characterize the posterior with a Markov Chain Monte Carlo algorithm. We introduce methods that summarize the high-dimensional uncertainty information and show how these summaries can be visualized in a meaningful way. Based on a clinical neurosurgical dataset, we demonstrate the importance that uncertainty information could have on neurosurgical decision making.