Five-bit teletypewriter codes ("Baudot", etc.)
As far back as the early 1900s (believe it or not) there were teletypewriters. They were intended to replace Morse-style telegraphy, directly printing hardcopy rather than requiring an operator to listen to the Morse code and transcribe the messages by hand.
The first successful such equipment was invented by a man named Baudot. He also invented the five-bit character code that his machines used. (See ["Baudot code"]) Later, a different five-bit (or "five-level") code was developed that made it easier to build a mechanical typewriter-like keyboard that would generate the code. The most successful such code was called ITA2.
These codes provided upper case only. They were primarily used for communication between machines like the Teletype Model 15, in networks like Telex, TWX, and Western Union's "Telegram" service.
These networks have almost completely faded away. Some ham radio operators still use Radio Teletype (RTTY) and they do still use the five-bit codes, partly due to tradition, partly for efficiency (they need fewer bits to send a character than the modern 8-bit codes). But this is most often done using computers as terminals, not ancient teletypewriters like the Model 15. A few of the die-hards do keep some of the old machines running.
Most old Teletype machines with three rows of keyboard keys (instead of the four that were common on typewriters) used these five-bit codes. They only needed three rows on the keyboard because numbers and special characters were shifted from the alphabetic keys. Two keys labeled "Figs" and, I believe, "Ltrs" sent the "shift in" and "shift out" codes. If you hear somebody talking about "three row" teletype machines, this is what they're referring to.
I wrote another answer about how teletypewriters in general, and the 5-level ("Baudot") machines in more detail, here.
The 6-bit TTS code
As for six-bit codes, the most common use (at least in terms of async serial communications) was probably the "TeleTypesetter" (TTS) code. I say this because virtually every newspaper that subscribed to a wire service (like AP or UPI) was equipped to receive it, usually with multiple feeds.
I'm going to say quite a bit about this because, while five-bit teletypewriter info is easy to find out about, there's very little out there about TTS.
The TTS code was a clear descendant of the five-bit codes like ITA2. Despite having six bits it still used "shift in" and "shift out" codes (like the Baudot code family did), permitting TTS to carry over 100 different glyphs (printable characters) and control commands. So it included upper and lower case alphabets, digits, a large assortment of special characters ("Wingdings" - far more than what you'd find on a typical typewriter), plus typesetting-oriented commands like "flush left" (which means "end a paragraph and justify the last line to the left"), center, and flush right.
In the wire room
In the old days, news wire services like AP and UPI (and smaller local ones, like City News Service) would send stories to their member newspapers via this code, over dedicated leased telephone lines. In each newspaper's "wire room" the copy would be printed on a keyboard-less TeleType Model 20 (so the editors could read and select the stories), and also fed to a "reperforator" (paper tape punch). There was one such pair of machines for each wire service the paper subscribed to.
Between each story the wire services would send a bunch of NULs (which would punch essentially blank tape, with only the feed holes), then a bunch of characters that would print as nonsense but would punch out the next-following Story ID on the tape in block letters, followed by more NULs. This made it relatively easy to find the section of tape that corresponded to the following printed copy. It also helped in identifying the correct direction and orientation for the paper tape, as the six-level tape was symmetric about the feed holes! But close inspection of the feed holes gave another way to tell the direction: They "led" the data holes slightly, so the back edge of a feed hole corresponded to the center line of the data holes it went with.
A new hire in the wire room would have the job of tearing off the hardcopy and the tape that went with it as it came out, filing the tape, and distributing the copy to the editors.
The tape for selected stories could then be fed directly to a Linotype machine equipped with a "Teletypesetter Operating Unit", which was a paper tape reader connected to a metal box that was placed on top of the Linotype's keyboard. The box was conceptually very simple: It had a solenoid for each of the Linotype's keys and it simply "pressed the keys" as the tape was read. The result, just as when a human was typing, was cast metal type that could be put on a press, inked up, and printed.
It was possible to edit the story before typesetting by using a TTS Teletype machine, tape reader, and tape punch ("reperforator"). The tape would be duplicated until the desired edit point was reached, then the operator could type additional text. Or to skip things, they would advance the original tape without copying it to the new tape.
A radio or TV station's news operation would have the model 20 Teletype, but no reperforator or Linotype. Copy from the Teletype would be torn off and handed to editors who turned it into the (generally much shorter) stories the anchors would read. In small stations the on-air news readers also did the writing. Eventually the wire services offered feeds already edited for radio or TV, used by smaller stations that didn't want to hire news copy editors.
The Teletype machines and paper tape punches ran nearly continuously, as stories were updated and reposted throughout the day. Ear protection was a good idea in the wire room! To this day, a few "all news" radio stations use the sound of one of those machines pounding out copy as background sound to their live on-air reporters.
TTS code and computer typesetting
In later years the incoming 6-bit signal was connected directly into a computer's serial port, stored on disk, and made available for review and editing via video terminals. After editing the computer would send the copy to a phototypesetter, resulting in nicely set type on photo paper.
(I spent a few years working for a newspaper with such equipment, based on HP 2100 minicomputers. We still had the reperforators but they were turned on only during the hour or so when we ran the nightly backups. Due to repetition on the feeds, almost all stories they wanted to use would be found already in the computer, but if need be they could find the tape for a missed story and read it via a paper tape reader attached to the computer. Incidentally our phototypesetters were made by Merganthaler, the same company that made Linotypes. The software system was called "Text II", from a company called Systems Development Corporation, based in Santa Monica.)
Virtually everything you ever saw printed in a newspaper with "AP", "UPI", or some other news "wire" service in the slug line came into the paper via systems like this. The local newspaper may, however, have edited the stories, sometimes significantly.
Around the late 70s, about the same time that I left the paper, AP was planning to offer a higher-speed network using eight-bit codes. So today, nearly 40 years later, I doubt that there's much if any of this six-bit TTS code left in use today - any more than there are five-bit ("three-row") teletype networks.
Well-maintained operating examples of TTS equipment are virtually nonexistent as there is virtually no hobbyist interest in it. One Teletype enthusiast site claims there is one operating example of a Teletype model 20.
Computers and six-bit codes
There were also many computers that used 6-bit character codes internally. Examples were the PDP-8 (a 12-bit machine, so two 6-bit characters could fit in a machine word), IBM 1401 and 7090, CDC 3000 and 6000 series (24-, 48-, and 60-bit words respectively), etc. These were not the same six-bit codes as TTS, and "shifting" was generally not used to expand the code set. Nor did they commonly use six-bit paper tape. Printers and punched cards of the day only supported upper case alphabets, so 64 different glyphs were enough. (See, for example, the Wikipedia article on the IBM 1401.) Some of them did, however, communicate with these six-bit codes over early modems.
IBM's System/360 set a de facto standard of 8-bit bytes (but using IBM's EBCDIC character code, which had very little uptake elsewhere), and several very successful mini-computers with 16-bit words (HP 2100, DG Nova, DEC PDP-11) started using 8-bit characters (generally using ASCII character codes) at around the same time. That pretty much ended the era of six-bit character codes.
The IBM 2741 and 1050 printing terminals were originally used with machines like the 7090 and the 1401, and these terminals used six-bit characters (at 134.5 bit/s, 1.5 stop bits). This was, however, a very different six-bit code from TTS, and in many cases different also from the character code used within the machines they connected to. (Of course.) Like the TTS code, though, they did have "shift in" and "shift out" commands, which corresponded literally to "shifting" the typewriter mechanism (ie rotating the ball 180 degrees). The 2741 and 1050 were based on the IBM Selectric mechanism; the Selectric typeballs of that time had 88 printable characters, already too many for six bits even before other controls are counted. (But not, alas, quite enough for the 94 printable glyphs of seven-bit ASCII.)
Later the same terminals were connected to System/360 and other eight-bit machines and the computers, or purpose-designed interfaces, had to do the code conversion. So if you happen to find a working 2741, yeah, you'll need that six-bit code to talk to it. :) You'll also need to be able to set your serial port to 134.5 bit/s.