This paper proposes and analyzes the use of effective and flexible fast-convolution (FC) implementation for single-carrier (SC) waveforms with very small roll-offs to reach high spectrum efficiency and significantly lower peak-to-average power ratio (PAPR) than corresponding multicarrier schemes. The generated waveforms are spectrally well-contained, with very small power leakage to adjacent frequencies, and are thus good candidates for opportunistic and heterogeneous spectrum use scenarios. The SC-waveforms are found to exhibit similar PAPR characteristics as the OFDM based SC-FDMA scheme. On the receiver side, fast-convolution filter bank (FC-FB) approach supports effective frequency-domain equalization with low complexity. It can be used for simultaneous processing of multiple SC channels with individually tunable bandwidths, center frequencies and timing offsets. This solution enables different terminals to operate with different roll-offs and hence with different PAPR. The approach supports well asynchronous multi-user operation, which is considered as an important element in 5G networks.