Being capable of enhancing the spectral efficiency (SE), faster-than-Nyquist (FTN) signaling is a promising approach for wireless communication systems. This paper investigates the doubly-selective (i.e., time- and frequency-selective) channel estimation and data detection of FTN signaling. We consider the intersymbol interference (ISI) resulting from both the FTN signaling and the frequency-selective channel and adopt an efficient frame structure with reduced overhead. We propose a novel channel estimation technique of FTN signaling based on the least sum of squared errors (LSSE) approach to estimate the complex channel coefficients at the pilot locations within the frame. In particular, we find the optimal pilot sequence that minimizes the mean square error (MSE) of the channel estimation. To address the time-selective nature of the channel, we use a low-complexity linear interpolation to track the complex channel coefficients at the data symbols locations within the frame. To detect the data symbols of FTN signaling, we adopt a turbo equalization technique based on a linear soft-input soft-output (SISO) minimum mean square error (MMSE) equalizer. Simulation results show that the MSE of the proposed FTN signaling channel estimation employing the designed optimal pilot sequence is lower than its counterpart designed for conventional Nyquist transmission. The bit error rate (BER) of the FTN signaling employing the proposed optimal pilot sequence shows improvement compared to the FTN signaling employing the conventional Nyquist pilot sequence. Additionally, for the same SE, the proposed FTN signaling channel estimation employing the designed optimal pilot sequence shows better performance when compared to competing techniques from the literature.