The Moral fibre of celebrities is \ are questionable.
I guess "The Moral fibre of celebrities" is a collective noun and should be treated as singular and, therefore, The Moral fibre of celebrities is questionable.
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It's likely the case that moral fibre is treated as a mass noun. It seems odd to me that you'd say something like this:
？ I don't approve of either of her moral fibres.
While fibres is countable, and we actually say he had good morals, we don't say he had only two good morals. In other words, the plural form of both morals and fibres is possible—but morals isn't used in an actually countable sense, nor would be the compound term moral fibres.
So, the question is do both of the following make sense?
✔ The moral fibre of celebrities is questionable.
？ The moral fibres of celebrities are questionable.
The singular version is undoubtedly correct. And if choosing between is and are as the word to follow the moral fibre of celebrities, it's irrelevant if moral fibre is a mass noun or not, because subject-verb agreement has is being used anyway.
From a purely syntactical point of view, the second sentence with are is the only version that could be used if the sentence starts with the moral fibres, but my inclination is to say that moral fibre is a mass noun and it would not be used in such a construction.