  In this paper we make a series of numerical experiments to support
Greenberg's $p$-rationality conjecture, we present a family of $p$-rational
biquadratic fields and we find new examples of $p$-rational multiquadratic
fields. In the case of multiquadratic and multicubic fields we show that the
conjecture is a consequence of the Cohen-Lenstra-Martinet heuristic and of the
conjecture of Hofmann and Zhang on the $p$-adic regulator, and we bring new
numerical data to support the extensions of these conjectures. We compare the
known algorithmic tools and propose some improvements.
