In this paper, we establish a framework for the spatio-temporal resource search problem. In a spatio-temporal resource search problem, a mobile agent tries to find a desired static resource in a network. The resources are statically located in the network. The task is to guide the agent through the network to find a resource with lowest possible cost. There are lots of applications in urban transportation systems that fit in this kind of setup. One example is a vehicle that searches for street parking in a neighborhood of a city. Another example is a taxicab that cruises in an area to find customers. In these examples, street parking spaces and taxicab customers are the static resources being searched. In this work, we develop two algorithms, namely the Random Walk Algorithm and the Prophet Algorithm, that serve as the lower and upper bounds on the performance of any reasonable algorithm that assumes some kind of information about the availability of the resources. Furthermore, when assuming that the resource-availability probabilities for each edge in the network are known to the agent, we develop the Probabilistic Algorithm using these probabilities. We conducted experiments for the three algorithms under different conditions, and the results show that using the Probabilistic Algorithm, the agent can perform much better than having no information at all, and surprisingly close to the performance upper bound.