Which spelling should I use: "grass roots" or "grassroots"?

Your question is certainly reasonable. I have noticed that dictionaries are somewhat slower than popular usage to pick up on spelling shifts that involve closing up hyphenated words or reducing two-word phrases to single words.

In the case of grassroots versus grass roots, Merriam-Webster's Eleventh Collegiate Dictionary (2003) lists grassroots as the preferred form for the adjective (with grassroot as an alternative) and grass roots as the preferred form for the noun. But as recently as the Ninth Collegiate (1983), MW offered no adjective form of the term, suggesting that it considered the normal form of the adjective to be either grass roots or grass-roots. That changed in the Tenth Collegiate (1993), when grassroots debuted as the preferred adjective form.

The Concise Oxford English Dictionary, tenth edition, revised (2002), meanwhile, echoes the Ninth Collegiate in offering only grass roots as the spelling for the noun form and no spelling for the adjective form.

For its part, The American Heritage Dictionary of the English Language, fifth edition (2010) lists grassroots as the only standard form for both the noun and the adjective. AHDEL came to this preference rather recently, however: in the fourth edition (2000), it listed grass roots as the noun form and grass-roots as the adjective form.

Clearly we are dealing with a term whose spelling is in flux—with Oxford holding to the traditional two-word or hyphenated spellings for the noun and adjective forms, Merriam-Webster opting for two words for the noun and one word (unhyphenated) for the adjective, and American Heritage endorsing the one-word form for both noun and adjective forms. My money, if I were a betting man, would be on the one-word form to triumph eventually, since the historical movement in spelling seems to favor it; but today we live in unsettled times, and the correct noun spelling of grass roots/grassroots depends on which authority you consult and how impressed you are by it.

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Update (January 12, 2021)

To provide a bit of real-world context, I offer the following Ngram chart of word/phrase frequency for the terms "grass roots" (blue line) versus "grassroots" (red line) versus "grass root" (green line) versus "grass root" (yellow line) for the period 1900–2019:

This comparison is not subtle, in that it doesn't distinguish between "grass roots" as a noun, an adjective, or a subject and verb (for example). Nevertheless, it tracks a quite striking rise in frequency of "grassroots" from essentially zero prior to 1940 to a plurality of usage by 1980 to a dominant position over the past three decades.

On the strength of these results, it's easy to see why AHDEL, fifth edition (2010) has embraced grassroots as the standard spelling and why MW began treating grassroots as the preferred spelling of the adjective form of the term in 1993.

As Edwin Ashworth notes in a comment beneath this answer, it is a truism that—to the extent that it takes a descriptivist approach to orthography—a dictionary will necessarily be somewhat behind times on current real-world spelling preferences in instances where those preferences are changing rapidly. Still, it bears observing that many dictionaries seem to exhibit a degree of inertial resistance to change when it comes to presenting words that they have spelled a certain way in the past, even when the tide of popular usage has clearly changed; and this inertial force, at some point, becomes difficult to distinguish from evidence-denying prescriptivism. After 40 years of real-world dominance of the form grassroots over the form grass roots, I think, a dictionary's continued insistence that grass roots is the only (or even the primary) acceptable spelling of the term qualifies as almost purely prescriptive.