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Unavoidable dead-ends are common in many probabilistic planning problems, e.g. when actions may fail or when operating under resource constraints. An important objective in such settings is MaxProb, determining the maximal probability with which the goal can be reached, and a policy achieving that probability. Yet algorithms for MaxProb probabilistic planning are severely underexplored, to the extent that there is scant evidence of what the empirical state of the art actually is. We close this gap with a comprehensive empirical analysis. We design and explore a large space of heuristic search algorithms, systematizing known algorithms and contributing several new algorithm variants. We consider MaxProb, as well as weaker objectives that we baptize AtLeastProb (requiring to achieve a given goal probabilty threshold) and ApproxProb (requiring to compute the maximum goal probability up to a given accuracy). We explore both the general case where there may be 0-reward cycles, and the practically relevant special case of acyclic planning, such as planning with a limited action-cost budget. We design suitable termination criteria, search algorithm variants, dead-end pruning methods using classical planning heuristics, and node selection strategies. We design a benchmark suite comprising more than 1000 instances adapted from the IPPC, resource-constrained planning, and simulated penetration testing. Our evaluation clarifies the state of the art, characterizes the behavior of a wide range of heuristic search algorithms, and demonstrates significant benefits of our new algorithm variants.