We explore the heating of the velocity distribution in the solar neighbourhood by stochastic spiral waves. Our investigation is based on direct numerical integration of initially circular test-particle orbits in the sheared sheet. We confirm the conclusion of other investigators that heating by spiral structure can explain the principal features of the age-velocity dispersion relation and other parameters of the velocity distribution in the solar neighbourhood. In addition, we find that heating by strong transient spirals can naturally explain the presence of small-scale structure in the velocity distribution (``moving groups''). Heating by spiral structure also explains why the stars in a single velocity-space moving group have a wide range of ages, a result which is difficult to understand in the traditional model that these structures result from inhomogeneous star formation. Thus we suggest that old moving groups arise from irregularities in the Galactic potential rather than irregularities in the star-formation rate.Comment: Accepted by MNRAS for publication; 20 pages, 11 figures, mn2e.cl