We present archival high-spatial-resolution VLA and VLBA data of the nuclei of seven of the nearest and brightest Seyfert galaxies in the Southern Hemisphere. At VLA resolution (~0.1arcsec), the nuclei of the Seyfert galaxies are unresolved, with the exception of MCG-5-23-16 and NGC7469, which show a core-jet structure. Three Seyfert nuclei are surrounded by diffuse radio emission related to star-forming regions. VLBA observations with parsec-scale resolution demonstrate that in MRK1239 the nucleus is clearly resolved into two components separated by ~30 pc, whereas the nucleus of NGC3783 is unresolved. Further comparison between VLA and VLBA data of these two sources shows that the flux density at parsec scales is only 20 per cent of that measured by the VLA. This suggests that the radio emission is not concentrated in a single central component, as in elliptical radio galaxies, and an additional low-surface-brightness component must be present. A comparison of Seyfert nuclei with different radio spectra shows that the `presence' of undetected flux on a milliarcsecond scale is common in steep-spectrum objects, whereas in flat-spectrum objects essentially all the radio emission is recovered. In the steep-spectrum objects, this `missing' flux probably results from non-thermal radiation related to active galactic nuclei, perhaps from a jet that is disrupted in Seyfert galaxies because of the denser environment of their spiral hosts.