A tightly correlated star formation rate-stellar mass relation of star forming galaxies, or star-forming sequence (SFS), is a key feature in galaxy property-space that is predicted by modern galaxy formation models. We present a flexible data-driven approach for identifying this SFS over a wide range of star formation rates and stellar masses using Gaussian mixture modeling (GMM). Using this method, we present a consistent comparison of the $z=0$ SFSs of central galaxies in the Illustris, EAGLE, and Mufasa hydrodynamic simulations and the Santa Cruz semi-analytic model (SC-SAM), alongside data from the Sloan Digital Sky Survey. We find, surprisingly, that the amplitude of the SFS varies by up to ${∼} 0.7\,\mathrm{dex}$ (factor of ${∼} 5$) among the simulations with power-law slopes range from $0.7$ to $1.2$. In addition to the SFS, our GMM method also identifies sub-components in the star formation rate-stellar mass relation corresponding to star-burst, transitioning, and quiescent sub-populations. The hydrodynamic simulations are similarly dominated by SFS and quiescent sub-populations unlike the SC-SAM, which predicts substantial fractions of transitioning and star-burst galaxies at stellar masses above and below $10^{10} M_⊙$, respectively. All of the simulations also produce an abundance of low-mass quiescent central galaxies in apparent tension with observations. These results illustrate that, even among models that well reproduce many observables of the galaxy population, the $z=0$ SFS and other sub-populations still show marked differences that can provide strong constraints on galaxy formation models.