We study the spreading of a wavepacket, evolving according to the Schrödinger equation, by means of the time-averaged moments of the position operator and discuss the following results. Lower bounds for the growth of these quantities on a power-scale may be obtained in terms of Hausdorff dimensional properties of spectral measures. Analogous upper bounds do not hold in general. In fact, transport may even be “faster” than the upper box counting dimension of the spectrum. Nevertheless, in one space dimension, upper bounds can be shown by proving lower bounds for transfer matrix growth at complex energies.