I more or less agree with Alcuin. From my point of view, most of the strange creatures of Middle-Earth were
inhabited by (mostly evil) spirits (in contrary to
being something themselves). The best it is shown on werewolves (as quoted by Alcuin above), but it in (some way) applies even on embodied Maiar as we imagine them: For example Sauron had many forms (he's shown as Annatar in the "nice outfit", as well as later the Necromancer, also as the "black warrior" when battling Elendil and Gil-Galad; and in the form of a werewolf and later then a vampire when fighting Beren, Lúthien and Huan - Lúthien here says that if he wouldn't give the power of Tol-in-Gaurhoth over to them, he'd be stripped of his body and he'll return before Morgoth and his "naked self" - hence, the true substantion of him - "will face eternal mockery"). Also, we can then read that Sauron was "mere hateful spirit" without the Ring (I think we all know these things). Powerful spirits like Sauron probably got more opportunities to "dress" themselves - as we read about the Valar, who chose to take the form which resembled their true personality - but the lesser spirits, like these who lived in werewolves, had in my opinion no choice: they were more likely something like
captured in the animal bodies (by Morgoth himself, most likely): they somehow "merged" with the animal body (for those who like objective demonstrations, I imagine it somehow like that there was animal body, animal soul - meaning mind, "brains" in the most general terms, and then the evil spirit itself). So in my opinion, besides Maiar as themselves, other creatures were most casually and most likely spirits who entered an animal body: werewolves and vampires (see above), great eagles (in contrary to some "normal birds flying around": the messengers of Manwë), even ents (Sil 2: "...after the Children awake... Yavanna's thought will rise as well, [Eru] will call spirits from far and they will come between kelvar and olvar, some will settle down there, and their rightful wrath shall be feared" - sorry again for the nonprecise quotation, I don't have the English original in possession). I also think the dragons (refer to Alcuin's post for more arguments on this) and barrow-wights are the ones who obviously fall into this cathegory: the wights of course were NOT something like revived spirits of the dead Dúnedain (which is complete nonsense, it is said even elves don't know where Men go after death, and when someone is dead, he is dead for good and goes to wherever Eru destinaded him to: the sole exception of this was Beren [Sil 12: "no one returned from there but Beren"], and I'd like to point out that Beren was still just waiting in Mandos and has not yet left the World. Of course, the spirits of Men leave unless kept in the Middle-Earth by force like the Dead Men of Dunharrow, but there was no need to keep there good-hearted descendants of Edain. Perhaps one or two evil princes of Cardolan will make sense, but not dozens of barrow-wights). It is said in LotR, that "the Witch-King called evil spirits from Rhudaur and Angmar" to infestate the barrows. So about these, I imagine it as spirit entering a (dead) body (bones, generally). Necessary to say, that the term "wight" meant simply "man" before, so whatever...? (I somehow cannot help but think of the possessed man from Mark 5 in association with the barrow-wights.)
Speculations about Kraken, the Giants, and Tom Bombadil I'll gladly leave to someone else
Although as Tolkien himself said, "a good story should have its mysteries". So that's about it from my side.