We select a sample of young passive galaxies from the Sloan Digital Sky Survey Data Release 7 in order to study the processes that quench star formation in the local universe. Quenched galaxies are identified based on the contribution of A-type stars to their observed (central) spectra and relative lack of ongoing star formation; we find that such systems account for roughly 2.5 per cent of all galaxies with log M_sun >= 9.5, and have a space density of ~2.2x10^-4 Mpc^-3. We show that quenched galaxies span a range of morphologies, but that visual classifications suggest they are predominantly early-type systems. Their visual early-type classification is supported by quantitative structural measurements Sersic indices that show a notable lack of disk-dominated galaxies, suggesting that any morphological transformation associated with galaxies transition from star-forming to passive--e.g. the formation of a stellar bulge--occurs contemporaneously with the decline of their star-formation activity. We show that there is no clear excess of optical AGN in quenched galaxies, suggesting that: i) AGN feedback is not associated with the majority of quenched systems or ii) that the observability of quenched galaxies is such that the quenching phase in general outlives any associated nuclear activity. Comparison with classical post-starburst galaxies shows that both populations show similar signatures of bulge growth, and we suggest that the defining characteristic of post-starburst galaxies is the efficiency of their bulge growth rather than a particular formation mechanism.
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