We consider various problems arising in optimal control of piecewise-deterministic Markov processes (PDMPs), with an emphasis on path planning problems where a randomly switching mode impacts the dynamics and optimization objective. We introduce two application-driven modeling frameworks, each formulated as a piecewise-deterministic path planning problem. The first framework models a vehicle that may experience breakdowns of varying severity at random times as it navigates a domain. We present an efficient iterative solver to recover the mode-dependent value functions and optimal policies for this problem. The second proposes a model of optimal foraging in a continuous domain while subject to predation. We present numerical experiments to demonstrate the impact of a forager's objective on the predicted optimal behavior. We close by studying a class of "occasionally observed" PDMPs, in which the planner is not notified when a mode switch occurs, but may occasionally have access to observations of the current mode. We state sufficient assumptions under which it is possible to represent the resulting belief over the modes as an explicit function of time. We present efficient dynamic programming algorithms for computing value functions and optimal policies for a variety of horizon types and observation schemes.