We study the unique, less-well understood problem of generating sparse adversarial samples simply by observing the score-based replies to model queries. Sparse attacks aim to discover a minimum number-the l0 bounded-perturbations to model inputs to craft adversarial examples and misguide model decisions. But, in contrast to query-based dense attack counterparts against black-box models, constructing sparse adversarial perturbations, even when models serve confidence score information to queries in a score-based setting, is non-trivial. Because, such an attack leads to i) an NP-hard problem; and ii) a non-differentiable search space. We develop the BruSLeAttack-a new, faster (more query-efficient) Bayesian algorithm for the problem. We conduct extensive attack evaluations including an attack demonstration against a Machine Learning as a Service (MLaaS) offering exemplified by Google Cloud Vision and robustness testing of adversarial training regimes and a recent defense against black-box attacks. The proposed attack scales to achieve state-of-the-art attack success rates and query efficiency on standard computer vision tasks such as ImageNet across different model architectures. Our artefacts and DIY attack samples are available on GitHub. Importantly, our work facilitates faster evaluation of model vulnerabilities and raises our vigilance on the safety, security and reliability of deployed systems.