Models of self-gravitating gas in the early stages of pressure-free collapse are compared for initial states that are equilibrium layers, cylinders, and ``Bonnor-Ebert'' spheres. For each geometrical case the density profile has an inner region of shallow slope surrounded by an outer region of steep slope, and the profile shape during early collapse remains similar to the profile shape of the initial equilibrium. The two-slope density structure divides the spherical collapse history into a starless infall phase and a protostellar accretion phase. The similarity of density profiles implies that Bonnor-Ebert fits to observed column density maps may not distinguish spherical cores from oblate or prolate cores and may not distinguish static cores from collapsing cores. The velocity profiles discriminate better than the density profiles between initial geometries and between collapse ages. The infall velocity generally has a subsonic maximum value, which is approximately equal to the initial velocity dispersion times the ratio of collapse age to central free-fall time. Observations of starless-core line profiles constrain collapse models. Collapse from initial states that are strongly condensed and slightly prolate is consistent with ``infall asymmetry'' observed around starless cores and is more consistent than collapse from initial states that are weakly condensed and/or oblate. Spherical models match observed inward speeds of 0.05-0.09 km/s over 0.1-0.2 pc if the collapse has a typical age of 0.3-0.5 free-fall times and if it began from a centrally condensed state that was not in stable equilibrium. In a collapsing core, optically thin line profiles should broaden and develop a two-peak structure, as seen in L1544, once the typical infall velocity approaches the molecular velocity dispersion or when the collapse age exceeds ∼0.4 free-fall times, for typical parameters, independent of depletion.