@Contrariwise This is a spectacular question with some fun subtlety. Because of the small size of index cards versus standard office paper, choosing a typewriter that has multiple affordances for holding onto the cards well is paramount. In my experience, not all of them do and many are much better than others.

Generally, you're looking for machines with card fingers near the typing point. These are usually one or (preferably) two pieces of metal which stick up right where the typeslugs hit the ribbon/paper. These help to hold the stiffer index card paper closer to the platen which makes both for a better ink imprint as well as preventing smudging above your typing line because the typeslug can double-strike the paper. Not having them can make your machine much louder while typing the first 3-4 lines which can be annoying.

On some later 60's models, typewriters replaced these card fingers with longer clear plastic card guides which do a much better job of holding the thicker index cards against the platen. The Olympia SM3 is a great example of this.

Some typewriters also included adjustable "paper fingers" which stick up typically in front of the platen, and like the card fingers help to hold the index card in place.

Almost all typewriters have a paper bail that holds the paper down from the top, but you'll want one that is more than just a simple bar. These usually won't work as well with stiffer cards as the paper bail isn't heavy enough to hold them down. You'll really want one with two or preferably three rubber rollers on the bail. Many of the ultra-portable typewriters (like the Smith-Corona Skyriter) of the 50s and 60s and cheaper typewriters into the 1970s are simple metal bars with no rollers. You really won't like these with index cards. Good rubber rollers on a paper bail can help the typewriter hold onto the index card and allow you to type right up to the very last line. Without them, you'll give up half an inch to an inch of typing space at the bottom of cards because the card will tend to lose alignment and may even fall out as it clears the platen and has nothing to hold it in place at the bottom.

Of my 20 machines, my personal favorites for index cards are (in order):
* 1950s Royal HH (a big heavy standard office machine); [manual]
* Smith-Corona 1950s Series 5 Silent and/or Silent Super which both have 3 rollers on the bail in addition to both card fingers and paper fingers; [manual]
* Olympia SM3 which has a large clear plastic card guard and rubber rollers on its bail; [manual]

I've added links to the manuals of these which include diagrams of some of the typewriter parts mentioned in case you're not familiar with the terms.

Depending on your use case(s), you'll also probably prefer a machine with an elite typeface (11-12 characters per inch) rather than the slightly more common pica typeface (10 characters per inch). The smaller elite typeface will allow you to type more words on a page which can be highly useful on smaller index cards where you'll quickly run out of room. I've heard rumors that some of the bigger Olympia standards like the SG1 and SG3 had even smaller typefaces as small as 17-18 characters per inch for index card work, but I've yet to encounter one in the wild personally.

Outside of index cards, if you're more of a notebook person, you could try some of the A5 notebook/binders which allow punched pages to be inserted and potentially re-arranged. This will give you more typing space on a page and free up some of the limitations on appropriate typewriter models. The Campus 20 ring A5 system comes to mind as a possibility here.

Let me know if you have any specific questions. I'm happy to help.

Incidentally I think there's an upcoming Micro.blog Analog (online) meetup on Saturday for discussing fountain pens, stationery, notebooks, etc. If that sounds interesting to you, ping @cygnoir or search her site for details to RSVP.