This paper considers the situation where users that experience high-mobility transmit data over frequency-selective channels, resulting in a doubly selective channel model (i.e., time- and frequency-selective channels) and this within the framework of Known Symbol Padding (KSP) transmission. KSP is a recently proposed block transmission technique where short sequences of known symbols acting as guard bands are inserted between successive blocks of data symbols. This paper proposes three novel channel estimation methods that allow for an accurate estimation of the time-varying transmission channel solely relying on the knowledge of the redundant symbols introduced by the KSP transmission scheme. The first method is a direct adaptive one while the others rely on a recently proposed model, the Basis Expansion Model (BEM), where the doubly selective channel is approximated with high accuracy using a limited number of complex exponentials. An important characteristic of the proposed methods is that they exploit all the received symbols that contain contributions from the training sequences and blindly filter out the contribution of the unknown surrounding data symbols. Besides these channel identification methods, the classical KSP equalizers are adapted to the context of doubly selective channels, which allows evaluation of the bit-error-rate (BER) performance of a KSP transmission system relying on the proposed channel estimation methods in the context of doubly selective channels. Simulation results show that KSP transmission is indeed a suitable transmission technique toward the delivery of high data rates to users experiencing a high mobility, when adapted KSP equalizers are used in combination with the proposed channel estimation methods.