We report a theoretical study of phase transition kinetics in confined two-dimensional systems, motivated by recent experimental results on the amorphous-to-crystalline transition in supported, thin amorphous water films [E.H.G. Backus, M.L. Grecea, A.W. Kleyn, and M. Bonn, Phys. Rev. Lett. (to be published)]. We generalize and extend existing theories to simultaneously describe the converted (crystalline) fractions in the bulk, at the sample-vacuum surface, and at the sample-support interface as a function of time. The general approach presented here results in expressions for the time-dependent converted bulk, surface, and interface fractions, for arbitrary desorption rate from the thin film, nucleation and growth rates and also includes finite nucleation grain size. The converted bulk, surface, and interface fractions are calculated for nucleation of the new phase occurring (i) in the bulk, (ii) at the support-sample interface, and (iii) at the sample surface (sample-vacuum interface), resulting in nine expressions. The results demonstrate the advantage of monitoring bulk, surface and interface fractions simultaneously to make definite statements regarding the location of the nucleation, and to reliably determine the values of the relevant crystallization parameters.