We have used the Rayner and Best smooth tests of goodness-of-fit to study the Gaussianity of the Very Small Array (VSA) data. These tests are designed to be sensitive to the presence of `smooth' deviations from a given distribution, and are applied to the data transformed into normalized signal-to-noise eigenmodes. In a previous work, they have been already adapted and applied to simulated observations of interferometric experiments. In this paper, we extend the practical implementation of the method to deal with mosaiced observations, by introducing the Arnoldi algorithm. This method permits us to solve large eigenvalue problems with low computational cost. Out of the 41 published VSA individual pointings dedicated to cosmological [cosmic microwave background (CMB)] observations, 37 are found to be consistent with Gaussianity, whereas four pointings show deviations from Gaussianity. In two of them, these deviations can be explained as residual systematic effects of a few visibility points which, when corrected, have a negligible impact on the angular power spectrum. The non-Gaussianity found in the other two (adjacent) pointings seems to be associated to a local deviation of the power spectrum of these fields with respect to the common power spectrum of the complete data set, at angular scales of the third acoustic peak (l = 700-900). No evidence of residual systematics is found in this case, and unsubtracted point sources are not a plausible explanation either. If those visibilities are removed, the differences of the new power spectrum with respect to the published one only affect three bins. A cosmological analysis based on this new VSA power spectrum alone shows no differences in the parameter constraints with respect to our published results, except for the physical baryon density, which decreases by 10 percent. Finally, the method has been also used to analyse the VSA observations in the Corona Borealis supercluster region. Our method finds a clear deviation (99.82 percent) with respect to Gaussianity in the second-order moment of the distribution, and which cannot be explained as systematic effects. A detailed study shows that the non-Gaussianity is produced in scales of l ~ 500, and that this deviation is intrinsic to the data (in the sense that cannot be explained in terms of a Gaussian field with a different power spectrum). This result is consistent with the Gaussianity studies in the Corona Borealis data presented in Génova-Santos et al. which show a strong decrement that cannot be explained as primordial CMB.