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Mobile crowdsensing systems aim to provide various novel applications by employing pervasive smartphones. A key factor to enable such systems is substantial participation of normal smartphone users, which requires effective incentive mechanisms. In this paper, we investigate incentive mechanisms for online scenarios, where users arrive and interact with a task requester in a random order, and they have preferences (e.g., photographing) or limits (e.g., travel distance) over the sensing tasks. In existing online mechanisms, the task requester has limited power in assigning tasks to the selected users, i.e., it has to pay for all of the tasks specified by the selected users, although some of these tasks are of little value. To accommodate this, we investigate a more flexible setting, where the requester can actively assign most valuable tasks to the selected users. We design two online incentive mechanisms motivated by a sampling-accepting process and weighted maximum matching. We prove that the designed mechanisms achieve computational efficiency, individual rationality, budget feasibility, truthfulness, consumer sovereignty, and constant competitiveness. By carrying out extensive experiments on two real-world geographical datasets, we demonstrate the practical applicability of the proposed mechanisms. [ABSTRACT FROM PUBLISHER]