We consider the classical multi-armed bandit problem and design simple-to-implement new policies that simultaneously enjoy two properties: worst-case optimality for the expected regret, and safety against heavy-tailed risk for the regret distribution. Recently, Fan and Glynn (2021) showed that information-theoretic optimized bandit policies as well as standard UCB policies suffer from some serious heavy-tailed risk; that is, the probability of incurring a linear regret slowly decays at a polynomial rate of $1/T$, as $T$ (the time horizon) increases. Inspired by their result, we further show that any policy that incurs an instance-dependent $O(\ln T)$ regret must incur a linear regret with probability $\Omega(\mathrm{poly}(1/T))$ and that the heavy-tailed risk actually exists for all "instance-dependent consistent" policies. Next, for the two-armed bandit setting, we provide a simple policy design that (i) has the worst-case optimality for the expected regret at order $\tilde O(\sqrt{T})$ and (ii) has the worst-case tail probability of incurring a linear regret decay at an exponential rate $\exp(-\Omega(\sqrt{T}))$. We further prove that this exponential decaying rate of the tail probability is optimal across all policies that have worst-case optimality for the expected regret. Finally, we generalize the policy design and analysis to the general setting with an arbitrary $K$ number of arms. We provide detailed characterization of the tail probability bound for any regret threshold under our policy design. Numerical experiments are conducted to illustrate the theoretical findings. Our results reveal insights on the incompatibility between consistency and light-tailed risk, whereas indicate that worst-case optimality on expected regret and light-tailed risk are compatible.