Monte Carlo simulations are used to investigate the asymptotic decay of the total pairwise correlation function h(r) for some model fluids. We determine the poles of the Fourier transform (h) over cap(q) from the direct correlation function c(r). The leading poles determine the ultimate, r -->infinity, decay of h(r). For the truncated and shifted Lennard-Jones fluid we calculate the Fisher-Widom (disorder) line in the temperature-density (T,rho) plane where the ultimate decay of rh(r) crosses over from monotonic (exponential) to exponentially damped oscillatory decay. This line lies close to that obtained in an earlier integral-equation [hypernetted chain-soft core mean spherical approximation (HMSA)] study. For states on the monotonic side of the disorder line, h(r) has a finite number of oscillations and we determine the boundaries which mark regions in the (T,rho) plane where h(r) has a given number of zeros using a random-phase approximation for c(r). In the case of the hard-sphere fluid, the ultimate decay of h(r) is oscillatory for all densities and we find that simulation results for the period and (exponential) decay length of the oscillations are in good overall agreement with those of Percus-Yevick theory, although there is some indication that systematic differences develop for high-density states rho*greater than or equal to 0.85.