Upcoming surveys such as the Vera C. Rubin Observatory Legacy Survey of Space and Time (LSST) will detect up to 10 million time-varying sources in the sky every night for ten years. This information will be transmitted in a continuous stream to brokers that will select the most promising events for a variety of science cases using machine learning algorithms. In this work, we study the benefits and challenges of Bayesian Neural Networks (BNNs) for this type of classification tasks. BNNs are found to be accurate classifiers which also provide additional information: they quantify the classification uncertainty which can be harnessed to analyse this upcoming data avalanche more efficiently.