Mumsnet was registered as a trademark in :
Class 9 - Electronic publications (downloadable); CDs; DVDs; Blu-ray discs; magnetic data carriers; recordings discs; data processing equipment and computers; photographic apparatus and instruments; apparatus for recording, transmission or reproduction of sound or images; computer software.
Class 16 - Paper, cardboard and goods made from these materials, not included in other classes; printed matter; photographs; stationery; books; magazines; publications.
Class 25 - Clothing, footwear, headgear.
Class 35 - Advertising services; business management services; business administration; office functions; retail services connected with the sale of household goods, clothing, footwear, nursery and baby goods, headgear, publications and printed matter, domestic electronic and electrical goods; promoting the goods and services of others via computer and communication networks.
Class 38 - Telecommunications services, namely, electronic transmission of data, messages and information; providing online forums for communication; providing online chat rooms and electronic bulletin boards.
Class 41 - Publication of electronic journals and web logs; education services; providing of training; entertainment services; sporting and cultural activities.
As of 2010.
I expect that if they wished, they could claim trademark infringement. Twitter should delete the account.