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?
You only have one subject in your sentence: "decades of research".
Research modifies decades, but decades is definitely plural, so the correct sentence is
Decades of research have shown
The same is true the other way:
A line of cars is standing on the road. A bag of apples costs $2.
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.
Adding to @oerkelens, research is almost irrelevant here because the the subject is 'Decades of x'
Decades of people
Decades of opinions
Decades of knowledge
It doesn't matter what there are decades of, have is still the appropriate word.
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.")
What I do in these cases is ask: has the research shown, or have the decades shown. You could make a case for both, but the case for research is stronger. So I'd recommend:
Decades of research has shown ...