When films of π-conjugated polymers are optically excited above a certain threshold intensity, then the emission spectrum acquires a multimode finely structured shape, which depends on the position of the excitation spot. We demonstrate that the power Fourier transform (PFT) of the emission spectrum exhibits a certain peak-like structure, which also depends on the excitation spot. Our intriguing observation is that averaging the individual PFTs does not lead to a structureless curve, but rather yields a series of distinct transform peaks. This suggests universality, namely that the underlying random resonators that are responsible for the laser emission from the π-conjugated polymer film are almost identical. We argue that the reason for such an universality is the large size of a typical resonator, which we determined from the PFT, as compared to the emission wavelength, λ. This fact is, in turn, a consequence of the large light mean free path, l10λ in the polymer film. This contrasts previous observations of random lasing in powders, where lλ. We develop a simple theory that explains the presence of peaks in the average PFT and predicts their shape. The results of the theory agree quantitatively with the data. To cite this article: R.C. Polson et al., C. R. Physique 3 (2002) 509-521.