The Doppler spread, or equivalently, the mobile speed, is a measure of the spectral dispersion of mobile fading channel. Accurate estimation of the mobile speed is of importance in wireless mobile applications which require the knowledge of the rate of channel variations. This paper exploits the inherent cyclostationarity of linearly modulated signals, transmitted through fading channels, to design robust blind and data-aided mobile speed estimators. Two categories of cyclic-correlation and cyclic-spectrum-based methods are developed. Extension to space-time speed estimation at the base station in macrocells is also provided. In comparison with the existing methods, the new estimators can be used without any need for pilot tones, and are robust to additive stationary noise or interference of any color or distribution. Unlike the conventional multi-antenna based method, the proposed space-time speed estimator does not assume the receiver noise to be spatially white. We also devised a suboptimal training sequence for pilot-symbol assisted methods, to reduce the estimation error. The performances of the proposed estimators are illustrated via extensive Monte Carlo simulations.