Which is the correct preposition to be used after 'VAIN'? Of or about? Does context make any difference?
What will we use in this sentence and why? She is vain _______ her achievements.
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This is a more interesting question than I thought.
For me, the answer is undoubtedly about: I would never say he is vain of (something), and if I heard it, I would think the speaker is not a native English speaker. But it looks as if I would be wrong in that judgment.
The OED (in an entry from 1916, not updated) defines this sense primarily as a description of a person, without an object. It does not have an example of the vanity being focussed on a particular aspect or property; but as a separate sub-entry it has "vain of" with the meaning you intend. The most recent example for that is from 1847.
Looking in the iWeb corpus, there are more vain of than I expected. It is hard to count precisely, because of the many examples where of happens to follow the phrase in vain, and also the many examples of the vain of which are clearly misspellings of the vein of. But even removing these, I am surprised to find more instances (176) of vain of than vain about (128).
I find a similar pattern in the GloWbE corpus. But when I look at COHA (The corpus of Historical American English), there is a clear pattern that vain of was more common until the 1930s, and vain about overtook it since then.
In summary: I would never use vain of, but it seems to be more common on the web, and is the only construction mentioned in the OED. Historically, it was more common than vain about until the 1930's, but vain about took over until the 1990s. On the web, things seem to have swung back.