This paper explores continuous-time and state-space optimal stopping problems from a reinforcement learning perspective. We begin by formulating the stopping problem using randomized stopping times, where the decision maker's control is represented by the probability of stopping within a given time-specifically, a bounded, non-decreasing, càdlàg control process. To encourage exploration and facilitate learning, we introduce a regularized version of the problem by penalizing the performance criterion with the cumulative residual entropy of the randomized stopping time. The regularized problem takes the form of an (n+1)-dimensional degenerate singular stochastic control with finite-fuel, where the regularized free boundary becomes the graph of a function mapping the state variable of the original stopping problem into the probability of stopping. We address this singular control problem through the dynamic programming principle, which enables us to identify the unique optimal exploratory strategy. Finally, we propose both model-based and model-free reinforcement learning algorithms tailored for exploratory optimal stopping problems. We establish policy improvement guarantees for the proposed algorithms. Moreover, the model-free method is of actor-critic type and it is scalable in high-dimensions under neural network parameterization.