Although radio-quiet quasars (RQQs) constitute greater than or equal to 90 per cent of optically identified quasar samples, their radio properties are only poorly understood. In this paper we present the results of a multi-frequency VLA study of 27 low-redshift RQQs. We detect radio emission from 20 objects, half of which are unresolved (less than or equal to 0.24 arcsec). In cases where significant structure can be resolved, doubler triple and linear radio sources on scales of a few kpc are found. The radio emission (typically) has a steep spectrum (alpha similar to 0.7, where S proportional to nu(-alpha)), and high brightness temperatures (T-B greater than or equal to 10(5) K) are measured in some of the radio components. The RQQs form a natural extension to the radio luminosity-absolute magnitude distribution of nearby Seyfert Is. We conclude that a significant fraction of the radio emission in RQQs originates in a compact nuclear source directly associated with the quasar. There are no significant differences between the radio properties of RQQs with elliptical hosts and those in disc galaxies within the current sample.