We investigate numerically the possibility of the soft-mode charge-density-wave (CDW) instability in a degenerate electron liquid with a rigid positive-charge background, on the basis of the dielectric formulation developed in previous papers. The estimated values for the critical density and the wave number for the onset of such an instability are rs85 and qc1.875qF. A physical account is given on the interrelationship between the Wigner transition, the violation of the Ferrell condition, and the CDW instability. It is shown through a numerical example that the "local approximation" for exchange and correlation can lead to manifestly inaccurate predictions on the onset of the CDW instability. © 1981 The American Physical Society.