Low-frequency current fluctuations are investigated over a bias range covering Ohmic, trap-filling, and space-charge-limited current regimes in polycrystalline polyacenes. The relative current noise power spectral density S(f) is constant in the Ohmic region, steeply increases at the trap-filling transition region, and decreases in the space-charge-limited-current region. The noise peak at the trap-filling transition is accounted for within a continuum percolation model. As the quasi-Fermi level crosses the trap level, intricate insulating paths nucleate within the Ohmic matrix, determining the onset of nonequilibrium conditions at the interface between the insulating and conducting phase. The noise peak is written in terms of the free and trapped charge carrier densities.