In addition to producing X-ray bursts and other time-variable phenomena, the nuclear burning of matter accreted on neutron stars provides a possible excitation mechanism for global oscillation modes of the star. Here we investigate the stability of both r- and g-modes in neutron stars using nonadiabatic pulsation calculations of steady state nuclear-burning surface envelopes on accreting neutron stars. We describe the pulsation spectrum and stability of the low-order r-modes for rotation periods between 50 and 500 ms. We find that the saturation of the CNO nuclear energy generation rate via fβ-decays plays a very important role in the pulsation stability of the thin, mass-accreting shells. Saturation effectively reduces the temperature sensitivity of the dominant nuclear reactions, producing a strong stabilizing effect on the pulsations. For example, we find that the fundamental r-modes are pulsationally stable when the CNO nuclear energy generation rate reaches this saturation. This occurs for the shell models with Mdot ≥ 0.002 MdotEdd. However, we also find that the fundamental r-modes of the shells with mass accretion rates as low as Mdot = 0.001 MdotEdd are excited in the unsaturated hydrogen-burning regime, with typical growth times of a few hours.We also investigate the spectrum and stability of the nonrotating g-mode oscillation spectrum. We find that these modes are also stabilized in the saturation regime; however, the low-degree g1-modes are also unstable in the unsaturated regime, with growth times on the order of hours. Due to the strong relationship between pulsational and thermal instability, revealed by the influence of saturation of the nuclear energy generation rate on the pulsational stability, the conditions under which mode excitation can occur based on the steady state calculations must be confirmed with more realistic time-dependent shell models. We will pursue these complexities in a subsequent paper. We also briefly discuss how some of these modes, if excited, might manifest themselves in the power spectrum of accreting neutron star systems. Upcoming X-ray missions, such as the X-Ray Timing Explorer, will have the collecting area and timing capability to probe the high-frequency power spectrum of neutron star systems with unprecedented sensitivity.