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There are currently just under 20 short vignettes gathered at stories.twitter.com. And these vignettes are indeed short: between 100 and 150 words in most cases, fitting for a service known for imposing a 140-character limit on all posts.
There's a heartwarming story about a man who saved his mother's bookstore and another about movie critic Roger Ebert's use of the service after he lost his voice. One user got a much-needed organ donation after sending this tweet: "sh*t, I need a kidney." Brands such as Burberry are represented as well.