Filaments of galaxies are known to stretch between galaxy clusters at all redshifts in a complex manner. In this Letter, we present an analysis of the frequency and distribution of intercluster galaxy filaments selected from the 2dF Galaxy Redshift Survey. Out of 805 cluster-cluster pairs, we find at least 40 per cent have bona fide filaments. We introduce a filament classification scheme and divide the filaments into several types according to their visual morphology: straight (lying on the cluster-cluster axis; 37 per cent), warped or curved (lying off the cluster-cluster axis; 33 per cent), sheets (planar configurations of galaxies; 3 per cent), uniform (1 per cent) and irregular (26 per cent). We find that straight filaments are more likely to reside between close cluster pairs and they become more curved with increasing cluster separation. This curving is toward a larger mass concentration in general. We also show that the more massive a cluster is, the more likely it is to have a larger number of filaments. Our results are found to be consistent with a cold dark matter cosmology.