This paper considers affine cyclic-prefixed block-based pilot-aided transmission (PAT) over the single-antenna doubly selective channel, where the channel is assumed to obey a complex-exponential basis expansion model. First, a tight lower bound on the mean-squared error (MSE) of pilot-aided channel estimates is derived, along with necessary and sufficient conditions on the pilot/data pattern that achieves this bound. From these conditions, novel minimum-MSE (MMSE) PAT schemes are proposed and upper/lower bounds on their ergodic achievable rates are derived. A pilot/data power allocation technique is also developed. A high-SNR asymptotic analysis of the ergodic achievable rate of affine MMSE-PAT is then performed which suggests that the channel's spreading parameters should be taken into account when choosing among affine MMSE-PAT schemes. Specifically, we establish that multicarrier MMSE-PAT achieves higher rates than single-carrier MMSE-PAT when the channel's delay-spread dominates its Doppler-spread, and vice versa.