Using linear non-adiabatic pulsation analysis, we explore the radial-mode (p-mode) stability of stars across a wide range of mass ( $0.2 \le M \le 50{\,\rm M_{{\odot }}}$ ), composition (0 ≤ X ≤ 0.7, Z = 0.001, 0.02), effective temperature (3000 ≤ T eff ≤ 40 000 K), and luminosity (0.01 ≤ L/M ≤ 100 000 solar units). We identify the instability boundaries associated with low- to high-order radial oscillations (0 ≤ n ≤ 16). The instability boundaries are a strong function of both composition and radial order (n). With decreasing hydrogen abundance we find that (i) the classical blue edge of the Cepheid instability strip shifts to higher effective temperature and luminosity, and (ii) high-order modes are more easily excited and small islands of high radial-order instability develop, some of which correspond with real stars. Driving in all cases is by the classical κ-mechanism and/or strange modes. We identify regions of parameter space where new classes of pulsating variable may, in future, be discovered. The majority of these are associated with reduced hydrogen abundance in the envelope; one has not been identified previously.