Phase-space descriptions are used to find qualitative features of the solutions of generalized scalar field cosmologies with arbitrary potentials and arbitrary couplings to matter. Previous results are summarized and new ones are presented as theorems, which include the previous ones as corollaries. Examples of these results are presented as well as counterexamples when the hypotheses of the theorems are not fulfilled. Potentials with small cosine-like corrections motivated by inflationary loop-quantum cosmology are discussed. Finally, the Hubble-normalized formulation for the FRW metric and for the Bianchi I metric is applied to a scalar field cosmology with a generalized harmonic potential, non-minimally coupled to matter, and the stability of the solutions is discussed.