A recent LCA study conducted by the Danish Environmental Protection Agency compared 14 types of bags and analyzed their impact in 14 categories. The study concludes that the most sustainable choice was to use LDPE bags - thick, glossy bags like the ones you get at department stores, and reuse them as trash bags. To compete with plastic in every LCA category, other alternatives such as regular cotton bags would need to be reused 7,100 times, and 20,000 times for organic cotton. Paper bags would need to be used 43 times to compete. The life-cycle study cites two reasons for this: plastic weighs less than paper or cotton, and carries more for the material it uses.
Access the full study here.