We present an analysis of the spectral properties of the extragalactic radio sources in the nearly complete Very Small Array (VSA) sample at 33GHz. Data from different surveys are used to study source spectra between 1.4 and 33GHz. We find that, in general, spectra cannot be well described by a single power law in the range of frequencies considered. In particular, most of the VSA sources that are steep between 1.4 and 5GHz, show a spectral flattening at ν > 5GHz. We identify 20 objects (19 per cent of the sample) clearly characterized by an upturn spectrum, i.e. a spectrum falling at low frequencies (ν <~ 5GHz) and inverted at higher frequencies. Spectra with high-frequency flattening or upturn shape are supposed to occur when the emission from the active galactic nucleus (AGN) compact core begins to dominate over the component from extended lobes. This picture fits well with the AGN unified scheme, for objects observed at intermediate viewing angles of the AGN jet. Finally, we discuss implications that this class of sources can have on future cosmic microwave background observations at high resolution.