It is generally assumed that galaxies are a bimodal population in both star formation and structure; star-forming galaxies are disks, while passive galaxies host large bulges or are entirely spheroidal. Here we test this scenario by presenting a full census of the kinematic morphologies of a volume-limited sample of galaxies in the local universe extracted from the MaNGA galaxy survey. We measure the integrated stellar line-of-sight velocity to velocity dispersion ratio (V/sigma) for 4574 galaxies in the stellar mass range 9.75<logM*[M_{sun}]<11.75. We show that at fixed stellar mass, the distribution of V/sigma is not bimodal, and that a simple separation between fast and slow rotators is oversimplistic. Fast rotators are a mixture of at least two populations, referred to here as dynamically cold disks and intermediate systems, with disks dominating in both total stellar mass and number. When considering star-forming and passive galaxies separately, the star-forming population is almost entirely made up of disks, while the passive population is mixed, implying an array of quenching mechanisms. Passive disks represent ~30% (both in number and mass) of passive galaxies, nearly a factor of two higher than that of slow rotators, reiterating that these are an important population for understanding galaxy quenching. These results paint a picture of a local universe dominated by disky galaxies, most of which become somewhat less rotation-supported upon or after quenching. While spheroids are present to a degree, they are certainly not the evolutionary end point for the majority of galaxies.