Abstract: We study the bandit problem where the underlying expected reward is a Bounded Mean Oscillation (BMO) function. BMO functions are allowed to be discontinuous and unbounded, and are useful in modeling signals with singularities in the domain. We develop a toolset for BMO bandits, and provide an algorithm that can achieve poly-log $\delta$-regret -- a regret measured against an arm that is optimal after removing a $\delta$-sized portion of the arm space.