We investigate the nature of the faintest radio sources detected in three Very Large Array surveys, to F(8.5 GHz) similar to 8 muJy. Using the Keck low-resolution imaging spectrograph in BRI and the near-infrared camera in K'(2.1 mum), we image 51 radio sources, and identify probable optical counterparts for 50. With low-resolution imaging spectroscopy, we successfully acquire new redshifts for 17 sources. Combining these with nine prior redshifts, we can then analyse a sample of 26 sources with spectroscopic redshifts. Based on this sample of 26, we find the largest contribution, about 60 per cent (15), to be from disc galaxies with high radio-to-optical ratios, indicating ongoing or very recent starbursts, at redshifts of z = 0-1. Most of these have large optical luminosities (M-B similar to -22 +/- 1), although two are sub-L* (M-B > -20). About 20 per cent (five) of the sample have relatively low ratios of radio-to-optical flux, consistent with normal, noninteracting galaxies, all at z < 0.4. Four are quasi-stellar objects (QSOs), of which one appears to be interacting and shows strong MgII absorption. Two others are very radio-luminous giant ellipticals and presumably contain active galactic nuclei. All of the 14 non-QSO galaxies with spectra show Hβ and [O II] λ3727 emission lines, with luminosities corresponding to star formation rates (SFRs) of the order of ∼10M(⊙) yr(-1). These SFRs are about an order of magnitude lower than estimated from the radio luminosities. This result may be explained if the galaxies are observed ∼10(7)-10(8) yr after a starburst and/or their regions of current star formation are heavily obscured by dust. Three of the galaxies show strong Hδ absorption and can be classed as post-starburst or 'e(a)' galaxies. The Keck spectra are of high enough resolution to provide approximate emission linewidths and rotation curves, as well as [O III]/Hβ ratios ('excitation'). Most of the galaxies (12/14) have high-amplitude kinematics (100-400 kms(-1)), implying large dynamical masses, and moderate excitation (∼0.51 ± 0.12), consistent with their morphological appearance as non-AGN, L ≥ L* spirals, of probable near-solar metallicity. The two with low optical luminosities appear to be low-mass galaxies with much higher excitations (∼2.8), and may be more similar to local H II galaxies than to normal spirals.