We present 42 rapidly evolving (time spent above half-maximum brightness $t_{1⁄2}<12$d) extragalactic transients from Phase I of the Zwicky Transient Facility (ZTF), of which 22 have spectroscopic classifications. This is one of the largest systematically selected samples of day-timescale transients, and the first with spectroscopic classifications. Most can be classified as core-collapse supernovae (SNe), and we identify several predominant subtypes: (1) subluminous Type IIb or Type Ib SNe; (2) luminous Type Ibn or hybrid IIn/Ibn SNe; and (3) radio-loud, short-duration luminous events similar to AT2018cow. We conclude that rates quoted in the literature for rapidly evolving extragalactic transients are dominated by the subluminous events (mostly Type IIb SNe). From our spectroscopic classifications and radio, X-ray, and millimeter-band upper limits, we are motivated to consider the AT2018cow-like objects a distinct class, and use ZTF’s systematic classification experiments to calculate that their rate does not exceed 0.1% of the local core-collapse SN rate, in agreement with previous work. By contrast, most other events are simply the extreme of a continuum of established SN types extending to ordinary timescales. The light curves of our objects are very similar to those of unclassified events in the literature, illustrating how spectroscopically classified samples of low-redshift objects in shallow surveys like ZTF can be used to photometrically classify larger numbers of events at higher redshift.