Until the 1960s, dinosaurs seemed to have more or less just appeared in the Late Triassic, with theropods and prosauropods (in the sense of "all them sauropodomorphs what ain't sauropods") recognized as present. What came before those theropods and prosauropods wasn't known, although there were a few guesses and extrapolations. One popular option of the time was that, technically speaking, there weren't really any basal dinosaurs because "dinosaurs" themselves were an artificial group "united" by some coincidental bits of anatomy related to being large land-living animals. This view is practically extinct, although I cannot say completely extinct. One of the great truths of humanity is that there is someone who will believe any proposition. One of the great truths of the Internet is that now you can find that person (or oftentimes, they will find you, if you are holding an opposing position). Other researchers drafted in various poorly known Triassic reptiles. The most enduring may have been Teratosaurus, which people who got into dinosaurs before the late 1980s will probably remember as a sort of megalosaur-like thing stalking the wilds of Late Triassic Europe. It was actually based on jaw material from a rauisuchian, with prosauropod skeletal remains misattributed to it. The misidentification of Teratosaurus, though illustrative, serves mostly as a lesson in the honesty of bonebeds. More recent work with Reveultosaurus, Shuvosaurus, and others shows that it can be darn hard to separate true early dinosaurs from the various wacky archosaurs of the Triassic if you've only got a few remains. The terrestrial Triassic still has more fools to make.
Of all the various bits and pieces put forward as early dinosaurs in the days before the 1960s, the only one that actually is both vaguely useful and does not easily slot into any of the known clades of true dinosaurs is Saltopus elginensis, described in 1910 by von Huene. It got to be in all the best dinosaur books as an archetype, overcoming the significant handicap of being a terrible specimen, which just goes to show that sometimes all you have to do to succeed is show up. The first useful basal dinosaurs to be described, Herrerasaurus and Ischisaurus, were described in 1963, followed by Staurikosaurus in 1970. They were followed by Lagosuchus, Lagerpeton, and Lewisuchus in 1971 and 1972, which were underappreciated at the time but eventually were shown to be dinosaurian cousins once we got that whole "unnatural Dinosauria" thing worked out of the collective scientific system. At the present, there are around 20 species of near-dinosaurs, from Dinosauromorpha to Dinosauria. This doesn't compete with, say, Titanosauria, but it's not bad for about 45 years of serious work. There's a little wiggle room built in depending on how charitable you feel toward Pseudolagosuchus, how you handle "Thecodontosaurus" alophos and borderline cases (e.g. Agnosphitys, Alwalkeria, Teyuwasu), and the occasional analysis that pulls Herrerasauridae or Eoraptor out of Saurischia. Some of them can be grouped as lagerpetids, diminutive bipeds, or as silesaurids, larger animals which could reach roughly the size and shape of Fred Flintstone's pet Dino. At least one of these, the namesake Silesaurus, was equipped with a little can-opener of bone at the tip of its lower jaw, perfect for being confused with the ornithischian predentary. A few others either don't slot comfortably into either group, or are poorly known (and I tend to the conservative when it comes to where I slot, which in this case mostly affects Lewisuchus/Pseudolagosuchus). The chart below lays them out with age and continent denoted. One thing to note is the abundance of species for South America and Africa, which not only suggests a Gondwanan origin but also provides a partial explanation for why these animals have only come to light in the past few decades: there haven't been a lot of paleontologists in those areas until recently.
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Another important thing to keep in mind is to avoid the trap of turning extinction and evolution into a morality play. "Near-dinosaurs" were not merely a sideline, or waiting hopefully to eventually evolve into dinosaurs, or a bunch of saps that got pushed out of the way by their cousins. They were their own creatures, diversifing into several lineages and living alongside their more famous cousins for twenty or so million years, at least. We currently have named examples from four continental landmasses, and it would hardly be surprising to add a few more landmasses, five to ten million years, or additional lineages.
|Dromomerom romeri, by Nobu Tamura (from https://commons.wikimedia.org/wiki/File:Dromomeron_BW.jpg). For whatever reason I've always found this restoration charming.|