From Singapore Airlines comes this answer:
[A] route between the Scottish Islands of Westray and Papa Westray with a scheduled flight time of two minutes. However, it was once flown in 58 seconds (it’s amazing what a strong tail wind can do to get you to your destination ahead of time). The flights are operated by Loganair for British Airways and utilise a Pilatus Islander prop plane. It is believed there is no beverage service, since even coffee takes three minutes to brew.
In other airline humour news, I was heartened to stumble across this Retirement of Our Random Fare Generator page from Alaska Airlines.