We present the results of a pilot JVLA project aimed at studying the bulk of the radio-emitting AGN population, unveiled by the NVSS/FIRST and SDSS surveys.We obtained A-array observations at the JVLA at 1.4, 4.5, and 7.5 GHz for 12 sources of the SDSS/NVSS sample. The radio maps reveal compact unresolved or slightly resolved radio structures on a scale of 1-3 kpc, with only one exception of a FRI/FRII source extended over $sim$40 kpc. We isolate the radio core component in most of them. The sample splits into two groups. Four sources have small black hole (BH) masses (mostly $sim$10$^{7}$ M$_{odot}$) and are hosted by blue galaxies, often showing evidence of a contamination from star formation to their radio emission and associated with radio-quiet AGN. The second group consists in seven radio-loud AGN, which live in red massive ($sim10^{11}$ M$_{odot}$) early-type galaxies, with large BH masses ($gtrsim$10$^{8}$ M$_{odot}$), and spectroscopically classified as Low Excitation Galaxies, all characteristics typical of FRI radio galaxies. They also lie on the correlation between radio core power and [O III] line luminosity defined by FRIs. However, they are more core dominated (by a factor of $sim$30) than FRIs and show a deficit of extended radio emission. We dub these sources FR0 to emphasize their lack of prominent extended radio emission, the single distinguishing feature with respect to FRIs. The differences in radio properties between FR0s and FRIs might be ascribed to an evolutionary effect, with the FR0 sources undergoing to rapid intermittency that prevents the growth of large scale structures. In our preferred scenario the lack of extended radio emission in FR0s is due to their smaller jet Lorentz $Gamma$ factor with respect to FRIs, causing possible instabilities and their premature disruption.[abridged]
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