In this paper, a cyclic-prefixed single-carrier (CPSC) transmission scheme with phase shift keying (PSK) signaling is presented for broadband wireless communications systems empowered by a reconfigurable intelligent surface (RIS). In the proposed CPSC-RIS, the RIS is configured according to the transmitted PSK symbols such that different cyclically delayed versions of the incident signal are created by the RIS to achieve multipath diversity. A practical and efficient channel estimator is developed for CPSC-RIS and the mean square error of the channel estimation is expressed in closed-form. We analyze the bit error rate (BER) performance of CPSC-RIS over frequency-selective Nakagami-$m$ fading channels. An upper bound on the BER is derived by assuming the maximum-likelihood detection. Furthermore, by resorting to the concept of index modulation (IM), we propose an extension of CPSC-RIS, termed CPSC-RIS-IM, which enhances the spectral efficiency. In addition to conventional constellation information of PSK symbols, CPSC-RIS-IM uses the full permutations of cyclic delays caused by the RIS to carry information. A sub-optimal receiver is designed for CPSC-RIS-IM to aim at low computational complexity. Our simulation results in terms of BER corroborate the performance analysis and the superiority of CPSC-RIS(-IM) over the conventional CPSC without an RIS and orthogonal frequency division multiplexing with an RIS.