I'm quite a latecomer to this, but thoroughly enjoying every minute of listening.
New Yorker Fiction Podcast
Writers reading other writers' short stories and then talking about it? Brilliant. However, some of the writers are better readers than others - for eg, Gary Shteyngart vs Mary Gaitskill? Shteyngart wins hands down. I'm afraid Gaitskill's reading of Vladimir Nabokov’s 'Symbols and Signs' made me very sleepy.
Selected Shorts Podcast
In these podcasts, actors such as Robert Sean Leonard and Cynthia Nixon read the short stories in front of a live audience.