The mechanical properties of nanocomposites based on styrene-butadiene rubber with fillers of carbon black, few-layer graphene and single-walled and multi-walled carbon nanotubes have been studied. Carbon nanotubes were subjected to preliminary purification according to a special technology developed by the authors using self-propagating high-temperature synthesis. It has been established that in elastomers filled with single-walled carbon nanotubes, the anisotropy of mechanical properties may appear, which is expressed in a significant increase of material rigidity and growth of dissipative tensile losses in the direction corresponding to the movement of vulcanized rubber compound during rolling. This effect was absent in composites filled with carbon black and few-layer graphene.