An Individual Survival Distribution (ISD) models a patient's personalized survival probability at all future time points. Previously, ISD models have been shown to produce accurate and personalized survival estimates (for example, time to relapse or to death) in several clinical applications. However, off-the-shelf neural-network-based ISD models are usually opaque models due to their limited support for meaningful feature selection and uncertainty estimation, which hinders their wide clinical adoption. Here, we introduce a Bayesian-neural-network-based ISD (BNN-ISD) model that produces accurate survival estimates but also quantifies the uncertainty in model's parameter estimation, which can be used to (1) rank the importance of the input features to support feature selection and (2) compute credible intervals around ISDs for clinicians to assess the model's confidence in its prediction. Our BNN-ISD model utilized sparsity-inducing priors to learn a sparse set of weights to enable feature selection. We provide empirical evidence, on 2 synthetic and 3 real-world clinical datasets, that BNN-ISD system can effectively select meaningful features and compute trustworthy credible intervals of the survival distribution for each patient. We observed that our approach accurately recovers feature importance in the synthetic datasets and selects meaningful features for the real-world clinical data as well, while also achieving state-of-the-art survival prediction performance. We also show that these credible regions can aid in clinical decision-making by providing a gauge of the uncertainty of the estimated ISD curves.