Decades of research has/have shown.
Should this sentence use a singular or plural verb?
decades is plural
research is murky
I'd be inclined to write 'has'. Should it be 'have' instead? Why or why not?
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Looking at the actual usage stats from the Corpus of Contemporary American English, singular agreement in this construction is possible, but rather uncommon:
decades of research has 2 decades of research have 14 decades of [nn*] has 4 decades of [nn*] have 60 centuries of [nn*] has 2 centuries of [nn*] have 14 months of [nn*] has 7 months of [nn*] have 21
Looking at the British National Corpus, again both variants are not unheard of, but the corpus size is too small to say anything decisively:
decades of [nn*] has 1 decades of [nn*] have 0 centuries of [nn*] has 0 centuries of [nn*] have 3 months of [nn*] has 2 months of [nn*] have 3
In conclusion, when writing for an international audience, I would favor plural. When writing for a local audience, I would favor whatever comes more naturally to me in my local dialect.
It depends on whether you're talking about the research or the decades. If you mean "Research which took decades", use the singular verb (research is a mass noun so it doesn't take plural verbs); if you mean "Decades, which were spent doing research", use the plural.
For example, "The decades of research were the happiest time of my life" but "Decades of research was needed to solve the problem." (Similarly, "tons of concrete was needed to fill the hole.")