Two more cases involving Twitter in the news today that are worth mentioning;
Firstly, one of the first ever suppression orders has been issued by a court in the UK that names Facebook and Twitter specifically as forms of media on which the publication of a particular person's name is suppressed - this seems to be partly a response to recent uses of Twitter to breach suppression orders obtained by celebrities, allegedly in many cases to prevent embarrassing personal information from being published in the media. This may be of limited success given the international nature of Twitter, but it certainly shows that social media networks are being taken seriously by the judiciary now days.
Second is an interesting case where someone is suing for the rights to control a Twitter feed. Essentially a teenager in California started a Twitter account which became wildly successful, attracting over 300,000 followers. Running the feed became a full time job, so he hired someone else to help out (it is implied he was making money off the Twitter feed somehow, though this is not explained in detail). The assistant / partner then little by little took over the feed until the original "owner" had been virtually excluded, prompting the lawsuit. To complicate matters, the assistant had indisputably done a lot of valuable promotion work, setting up an associated website and YouTube channel, and expanding the follower base from ~350,000 to 1.9 million followers, as well as claiming that the original "owner" of the feed had barely made a tweet since hiring the assistant to help out.
So the question mainly rests on who owns the trademark to the "brand" constituted by the Twitter feed and its associated image, name etc. However there is some interesting legal analysis about the copyright status of tweets in general, with legal opinion that (i) tweets are so short by nature that they struggle to reach the length at which a work would become copyrighted, and (ii) because of the importance of the "retweet" function built into Twitter, this implies that the nature of the service involves an inherent waiver of normal copyright rights, in that retweets would otherwise be breach of copyright, and the fact that they are so central to the Twitter service means that normal copyright conventions must therefore not apply.