This paper investigates several new strategies for the allocation of radio resources (bandwidth and transmission power) using a non-orthogonal multiple access (NOMA) scheme with successive interference cancellation (SIC) in a cellular downlink system. In non-orthogonal access with SIC, the same subband is allocated to multiple users, which necessitates more elaborate multiuser scheduling and subband assignment techniques, compared to orthogonal multiplexing. While taking into account various design issues, we propose and compare several optimum and suboptimum power allocation schemes, which are jointly implemented with multiple user scheduling strategies. Besides, a minimization of the total amount of used bandwidth is targeted. Also, to increase the total achieved system throughput, a hybrid Orthogonal-Non orthogonal scheme is introduced. This hybrid scheme enables a dynamic switching to orthogonal signaling whenever the non-orthogonal cohabitation in the power domain does not improve the achieved data rate per subband. Extensive simulation results show that the proposed strategies for resource allocation can improve both the spectral efficiency and the cell-edge user throughput, especially when compared to previous schemes employing either orthogonal signaling or NOMA with static inter-subband power allocation. They also prove to be robust in the context of crowded areas.