There is no published literature on this particular case of lexical variation.
Sofa is the preferred variant throughout the UK, followed by settee, and finally couch. However, there is variation in the degree to which this variant is preferred in each of these major regions. Sofa is most predominant in the South where it is used by 79% of speakers, whereas in the North we find a much more even distribution between the three variants, with sofa comparatively less popular as it is used by only 42% of speakers here. Couch shows a high concentration in the North West, particularly in Merseyside.