We have tested Very Small Array (VSA) observations of three regions of sky for the presence of non-Gaussianity, using high-order cumulants, Minkowski functionals, a wavelet-based test and a Bayesian joint power spectrum/non-Gaussianity analysis. We find the data from two regions to be consistent with Gaussianity. In the third region, we obtain a 96.7 per cent detection of non-Gaussianity using the wavelet test. We perform simulations to characterize the tests, and conclude that this is consistent with expected residual point source contamination. There is therefore no evidence that this detection is of cosmological origin. Our simulations show that the tests would be sensitive to any residual point sources above the data source subtraction level of 20 mJy. The tests are also sensitive to cosmic string networks at an rms fluctuation level of 105 μK (i.e. equivalent to the best-fitting observed value). They are not sensitive to string-induced fluctuations if an equal rms of Gaussian cold dark matter fluctuations is added, thereby reducing the rms fluctuations due to the strings network to 74 μK. We especially highlight the usefulness of non-Gaussianity testing in eliminating systematic effects from our data.