On June 8, 1959, the US Navy submarine USS Barbero launched a Regulus cruise missile from off the southeastern US coastline. Its target was a naval base near Jacksonville, Florida. Courtesy of 150,000 Newtons of thrust delivered by solid-propellant boosters, it made the trip in about 22 minutes, hitting its target with astounding precision.
Jacksonville wasn't incinerated in a fireball because, unlike most of its kin, the Regulus was stripped of its nuclear warhead. In its place were two canisters of US mail. See, the USPS had established an official post office on the submarine, and, prior to its departure from Norfolk, had passed on 3,000 pieces of mail, all of them ceremonial postcards addressed to various US officials, including President Dwight Eisenhower.
"This peacetime employment of a guided missile for the important and practical purpose of carrying mail, is the first known official use of missiles by any Post Office Department of any nation," said United States Postmaster General Arthur E Summerfield. "Before man reaches the moon, mail will be delivered within hours from New York to California, to Britain, to India or Australia by guided missiles. We stand on the threshold of rocket mail."
It was a stunt, of course. Save for the single test flight, rocket mail wasn't a thing—though various goofballs and tinkerers had given it a go in the 1920s and '30s, with little success—and by 1959, air mail had already proven itself to be the future of correspondence, at least for the next few decades. Whereas mail only a few decades before could take weeks to make the trip from the US to Europe, it could now be sent is as little as a day with no rocketry required. There's also the whole matter of blowing a million-dollar rocket on a backpack's worth of letters. People complain that mail is inefficient now.
The truth is more interesting, besides. The mail rocket was a stunt, but it was also a flex. These were the early days of the Cold War and blasting a long-range nuclear-capable rocket across the southeastern United States was a chance to show off its terrifying speed and precision. The Regulus had a range of 600 miles and could be delivered essentially to your doorstep.
So, no, rocket mail never really happened. Full-on rockets aren't particularly efficient for anything not destined for space or mass-destruction.
Still, some have kept the dream alive. XCOR Aerospace managed to develop a rocket-plane called the EZ-Rocket, albeit mostly to demonstrate its reusable rocket technology and to break some records. Test pilot Dick Rutan made one nine-minute flight from Mojave, California to California City, California. Its cargo: US mail. The EZ-Rocket was planned to be retired shortly thereafter.