The desire for power in men was clearly showcased a fair few times throughout the Tolkien books/movies, especially noticeable as a major flaw in men as a race with Gondor seeking the ring to use its power and save their plight.
The greed of dwarves is a main theme in the hobbit, and Thorin's story arc. They even have an illness for it "Dragon sickness".
Elves are not portrayed quite as flawed as dwarves and men, but if you look (not too much) deeper, you can notice their pride and vanity being shown occasionally. For some examples I would point to their strife with the stubborn (I suppose that's another flaw) dwarves and how elves refuse to acknowledge that they are at all responsible for the tensions between them. When the Fellowship arrives at the doors of Moria, Galdalf explains that there was friendship between the Dwarves of Moria and the Elves of Hollin.
Those were happier days, when there was still close friendship at times between folk of different race, even between Dwarves and Elves.’
‘It was not the fault of the Dwarves that the friendship waned,’ said Gimli.
‘I have not heard that it was the fault of the Elves,’ said Legolas.
‘I have heard both,’ said Gandalf; ‘and I will not give judgement now. But I beg you two, Legolas and Gimli, at least to be friends, and to help me.
The Lord of the Rings Book Two, Chapter 4: A Journey in the Dark
Page 303 (Single volume 50th Anniversary Edition)
In general, Tolkien seems to love using flaws in his characters, and there are tons of examples in every direction, some character specific, and some applying to the whole race.