Well, the Modern Language Association is keeping up with citations and now has announced their official format for citing a Tweet.
If you think that Twitter is just for silly stuff, I guess you must have missed news stories from the "Arab Spring" to the earthquake in Japan to reactions to Bank of America fees to the death of Steve Jobs or Whitney Houston. News often breaks first on social networks like Twitter.
The basic format looks like this
Last Name, First Name. (User name). "The entire tweet." Date, Time. Tweet
There is no URL - maybe because (unfortunately) Twitter is really difficult to search for posts that go back more than a few days.
Here are two examples:
Ronkowitz, Ken (ronkowitz). "Everything That Happens In One Day On The Internet http://t.co/wxLu6Lvr" 6 March 2012, 8:26 p.m. Tweet.
poetsonline.org (poetsonline). "The Poet & Writer Evening News is out! bit.ly/e5lWkC" 21 Feb 12 2012, 6:41 p.m. Tweet.
Here is the actual MLA instruction for citing:
Begin the entry in the works-cited list with the author's real name and, in parentheses, user name, if both are known and they differ. If only the user name is known, give it alone. Next provide the entire text of the tweet in quotation marks, without changing the capitalization. Conclude the entry with the date and time of the message (using the reader's time zone)and the medium of publication (Tweet).
I am not sure if the American Psychological Association (APA) has it in their guide yet.
UPDATE: For APA information, take a look at this post (Hat tip to Dr. Lonna Murphy)