We review the current status of heavy quarkonium production phenomenology based on nonrelativistic effective field theories, focusing on spintriplet S -wave states such as J/ψ, ψ(2S ), and ϒ. We present some representative examples for heavy quarkonium production mechanisms proposed in the literature, which vary significantly depending on the choice of data employed in analyses. We then discuss the rôle of polarization in discriminating between the different possible scenarios for quarkonium production. Other observables that may be useful in pinpointing the production mechanism are also introduced, such as the ηc production, associated production of J/ψ plus a gauge boson, and J/ψ production at the Electron-Ion Collider.