As a first follow-up to the post on potentially iconic races, I wanted to draw some generalizations about the races across all listed themes. The basic categories are Exotics, Beasts, Demi-Beasts, and Supernaturals, plus (maybe) one other mentioned in the comments: Homunculi/Constructs (although these might be considered a subtype of Supernatural, along with Fae, Accursed, and Elementals.) Each of these categories could be taken as a broad iconic "race", but individual examples of each category tend to be too specific to a theme to be broadly applicable. For example, anyone really delving into classical mythology and legend will find several kinds of Exotic, but only two (Amazons and Giants) have what I would call broad cultural recognition to those who aren't "Greek and Roman Mythology Nerds". Similarly, the classical Demi-Beasts that would make individually recognizable races to a broad audience would be Centaurs, Satyrs, Minotaurs, and maybe Harpies and Sphinxes.
What this means is that, if you want a race that a typical player could look at and say "I could play that," it's those races, perhaps mixed with a few other iconics from other themes (lizard/snake/reptile people from swords & sorcery/modern conspiracy and UFO theory, for example.) These have the rudimentary suggestion of typical behavior and culture. Other races might be possible, but it's better to allow those as unique or solitary examples, so that you don't have to worry about the cultural details of the race: the hawk-man either shares human cultural and acts like a man who just happens to have a hawk's head/wings, or the player makes up all the details of the hawk-man race in bits and pieces during play.
A second general point to make is that the more mundane Exotics with sketchy cultural details make better PC races than the others, although the general trend in RPG supplements is to push towards the Demi-Beast/Supernatural end and pile on cultural details. If you look at Tolkien, what he did for his standard "races" was take Fae beings (elf, dwarf, "hob", goblin) and shift them towards the more mundane end, turning them into Exotics. He also elaborated cultures for three of the races, pretty much turning them into actual tribes, and left the more powerful and extreme creatures in reserve as monsters or rescuers to encounter. He added elaborate details because he was writing books, so he had the space to do so; gamers need something simpler that you can just jump right into, since one person's vision of elven society isn't going to be iconic, while the most general concepts will be.
A third general point: I separated the races by theme because it's not just individual races that are iconic, but also the palette of races. Swords & Sorcery works better if most of your "races" are just humans with cultural differences and the occasional unique wild half-man; this is because S&S paints a picture of a very human-dominated world with mysterious inhuman mysteries, and having too many powerful supernatural or monstrous races roaming about dominating the landscape inhibits that theme. You *can* allow Fae spellcasters to slip in for an Arthurian-themed campaign, but then tossing in several Greek Demi-Beasts puts too much strain on the campaign, at least in the broadly-playable sense. An individual group could go in for a campaign of knights, priests, elves, centaurs, and medusae, maybe, but other groups would reject that as unappealing.
I can't prove this, but I suspect that the more varied your racial palette, the fewer races there needs to be. Specifically, I'm thinking that the further up the Exotic -> Beast -> Demi-Beast -> Supernatural scale you go for core, defined races, the more restricted the racial palette needs to be to create a campaign theme that feels coherent. You can have a campaign world where medusae and their gargoyle slaves vie for control with elves and humans, but adding a bunch of other Demi-Beast or Supernatural civilizations that all trade or war with each other just feels like a mess. This doesn't mean that players are necessarily restricted to only those races; they can still play as unique or solitary individuals.They stand out and become legends.