We analyse the kinematics of ∼400 000 stars that lie within ∼2 kpc of the Sun and have spectra measured in the Radial Velocity Experiment. We decompose the sample into hot and cold dwarfs, red-clump and non-clump giants. The kinematics of the clump giants are consistent with being identical with those of the giants as a whole. Without binning the data we fit Gaussian velocity ellipsoids to the meridional-plane components of velocity of each star class and give formulae from which the shape and orientation of the velocity ellipsoid can be determined at any location. The data are consistent with the giants and the cool dwarfs sharing the same velocity ellipsoids, which have vertical velocity dispersion rising from 21 km s−1 in the plane to ∼55 km s−1 at |z| = 2 kpc and radial velocity dispersion rising from 37 km s−1 to 82 km s−1 in the same interval. At (R, z), the longest axis of one of these velocity ellipsoids is inclined to the Galactic plane by an angle ∼0.8 arctan(z/R). We use a novel formula to obtain precise fits to the highly non-Gaussian distributions of vϕ components in eight bins in the (R, z) plane. We compare the observed velocity distributions with the predictions of a published dynamical model fitted to the velocities of stars that lie within ∼150 pc of the Sun and star counts towards the Galactic pole. The predictions for the vz distributions are exceptionally successful. The model's predictions for vϕ are successful except for the hot dwarfs, and its predictions for vr fail significantly only for giants that lie far from the plane. If distances to the model's stars are overestimated by 20 per cent, the predicted distributions of vr and vz components become skew, and far from the plane broader. The broadening significantly improves the fits to the data. The ability of the dynamical model to give such a good account of a large body of data to which it was not fitted inspires confidence in the fundamental correctness of the assumed, disc-dominated, gravitational potential.