A sensor network often consists of a large number of sensing devices of different types. Upon deployment in the field, these sensing devices form an ad hoc network using wireless links or cables to communicate with each other. Sensor networks are increasingly used to support emergency responders in the field usually requiring many sensing tasks to be supported at the same time. By a sensing task we mean any job that requires some amount of sensing resources to be accomplished such as localizing persons in need of help or detecting an event. Tasks might share the usage of a sensor, but more often compete to exclusively control it because of the limited number of sensors and overlapping needs with other tasks. Sensors are in fact scarce and in high demand. In such cases, it might not be possible to satisfy the requirements of all tasks using available sensors. Therefore, the fundamental question to answer is: “Which sensor should be allocated to which task?", which summarizes the Multi-Sensor Task Allocation (MSTA) problem. We focus on a particular MSTA instance where the environment does not provide enough information to plan for future allocations constraining us to perform instantaneous allocation. We look at this problem in both static setting, where all task requests from emergency responders arrive at once, and dynamic setting, where tasks arrive and depart over time. We provide novel solutions based on centralized and distributed approaches. We evaluate their performance using mainly simulations on randomly generated problem instances; moreover, for the dynamic setting, we consider also feasibility of deploying part of the distributed allocation system on user mobile devices. Our solutions scale well with different number of task requests and manage to improve the utility of the network, prioritizing the most important tasks.