A one-dimensional Kronig-Penney model is utilized to examine the influence of the lattice potential on the total energy distribution of electrons field emitted from clean and adsorbate-covered metal surfaces. In the case of clean surfaces, we demonstrate that the lattice potential influences the energy distributions primarily near the gaps in the (bulk) energy-band structure. Even when the effects of the lattice potential are strong, however, the value of the work function may be extracted to within an accuracy of a few percent from the energy distribution and plots of the emission current as a function of the inverse of the electric field strength. We also examine the validity of three common approximations, the free-electron model, the effective-mass-method and the tight-binding approximation, used to evaluate energy distributions. Of these, only the tight-binding approximation is valid over wide ranges of model parameters, and then only if the model parameters are obtained empirically rather than from nearest-neighbor overlap integrals. The effective-mass method almost always fails to provide an accurate approximation scheme. Our study of the influence of the lattice potential on emission through adsorbed species indicates that major deviations from the lineshapes predicted using the free-electron model can occur in the cases of narrow bulk energy bands or strong adsorbate-lattice interactions. The influence of the adsorbate on the energy distribution is described by an enhancement factor defined to be the ratio of the energy distributions with and without the adsorbate. Using the free-electron model for the metal, the enhancement factor exhibits a simple peak near the binding energy of the adsorbate. The lattice potential can cause substantial shifts of the energy of this peak as well as the occurrence of subsidiary maxima near the extrema energies of the allowed energy bands of the bulk solid. The combined effects of an adsorbate resonance and the bulk lattice potential provide interpretations of both experimental data for Ca adsorption on W (100) and of experimentally observed trends in the total energy distribution upon adsorption of various species onto the same surface. This interpretation does not require consideration of field emission from surface states.