'Decades of research {has/have} shown.'

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.