What’s the point of such a flight? Just from looking at the map, it looks like there might be 100 people on Papa Westray, maybe 1000 on Westray, with several ferry routes. The only rationale I can think of is maybe if the guy who owns most of Papa Westray lives on Westray, flies back and forth on a regular basis, and keeps the plane running for his tennants and neighbors.
Postal services for those islands?
There are nineteen airports in Indiana.
And while it didn’t make the list, Pam’s Place Airport deserves mention.
Found this from Era Aviation:
Aniak, AK to Chuathbaluk, AK which is just short of 16km away, but has no roads connecting them.
Looks like Era offers a daily (except Sundays) flight at 12:10pm from Aniak, and the flight back to Aniak departs at 12:25pm. Flight time is 10 minutes and the trip costs $90 each way.
It’s so the airline doesn’t have to do a separate flight to each island. AFAIK the main purpose is to go to and from Kirkwall (and from there on to wherever), not between the islands.
I flew from Tahiti to the island of Moorea, 7 minutes.
The shortest I can find in the continental USA is from Baltimore (BWI) to Philadelphia (PHL) US Airways flight 4601, 37minutes.
Silver Airways (operating for United Express) has flights between Bradford, PA and Jamestown, NY. That’s only about 40 miles. While you can book just that segment, the flight is operating as a stopping service to/from Cleveland. Bradford is losing its Essential Air Service subsidy, though, so it won’t have commercial flights for much longer.
Are you talking about flight time or total time? Baltimore to Philadelphia is a short air route but Denver to Colorado Springs is even shorter by air. I have taken both of them by chance in the past year. The Denver to Colorado Springs route is only 14 - 20 minutes depending on the aircraft and the routing that you get for a total distance of about 80 miles. It isn’t the shortest in the world but it is notable because it route that is routinely used between two large airports. The total time is about 40 minutes and most of that is spent taxiing.
Are you talking about flight time or total time? Baltimore to Philadelphia is a short air route but Denver to Colorado Springs is even shorter by air. I have taken both of them by chance in the past year. The Denver to Colorado Springs route is only 14 - 20 minutes depending on the aircraft and the routing that you get for a total distance of about 80 miles. It isn’t the shortest in the world but it is notable because it route that is routinely used between two large airports. The total time is about 40 minutes and most of that is spent taxiing.
Reagan National to Dulles would be shorter; while I doubt it’s offered as a destination, many flights out of National stop at Dulles on their way to a real destination.
The shortest flight I’ve been on was from Vegas to the Grand Canyon. Unfortunately there was an enormous dust storm approaching so we taxied, then went back and disembarked. 
I doubt I’ll be honeymooning in the States again, any time soon.
Although no longer in business (last flights were in 1986), Island Airlines provided scheduled flights from Port Clinton, Michigan to a series of islands in Lake Erie.
These flights started at the airport in Port Clinton and landed at Kelleys, South Bass, Middle Bass, and North Bass islands. It was dubbed the “World’s Shortest Airline” because the round-trip flight was 17 miles and took just 45 minutes.
The really neat thing about these flights was that they exclusively used Ford Trimotors. As the last Trimotor was built in 1933, this meant that the airplane had a commercial lifespan of 53 years,
I’ve flown that route many times as a connecting flight. I figured that SBY to BWI (which I’ve flown many times, but not in 8-9 years) was pretty short, but it looks like they only go to PHL and CLT these days.
These planes fly at different speeds, right? So that might be needed comparing proximity on land.
Eilat-to-Tel Aviv is pretty quick, I suppose, but I’ve never taken it.
The shortest I’ve ever flown was Vancouver BC to Victoria BC. It took about 15 minutes.
People’s Viennaline is making history by launching the world’s shortest international flight. The flight, from St. Gallen, Switzerland (ACH) to Friedrichshafen, Germany (FDH), covers a distance of 13 miles and lasts just eight minutes in the air (though it’s blocked for 20).
Read more: Austrian Carrier Has World's Shortest International Flight - The Points Guy
Daily Mail story has more and summary of other longest and shortest flights–which perhaps some more enterprising poster will compare to those already in this thread for completeness/correcti on either part. Airline announces the world's shortest international flight, with the 13-mile journey taking just EIGHT MINUTES | Daily Mail Online
I flew the opposite way last month: Moorea to Tahiti on Air Tahiti. The flight took 10 minutes. That was the shortest regularly scheduled flight I have ever taken.
What’s the point of these 2-15 minutes flights? Aside from areas in very rough terrain, isn’t it simpler and cheaper to make a road or go by boat?
The most obvious is that you have arrived at the airport by another flight and are simply connecting for a final destination (or the reverse.) They may also be stopping points on a cycle of airports the plane runs around rather like a bus service.
Boat can easily involve a whole lot more pain time and expense than a plane. Even a small plane can cover a lot of ground in 15 minutes.