What's the Preferred Way to Write the Abbreviation for United States?

Even though the question of how to abbreviate the United States seems straightforward, as it happens, there's more than one preferred way to write it. But before getting into that, let's get it out of the way first to note that if your usage of the country name is a noun, spell it out rather than abbreviating it. If it's an adjective, then how to do so becomes the question. (And obviously, if you're writing something formal, you'll want to follow the style guide to which you're assigned to adhere.)

Use Periods

In general, newspaper style guides in the United States (in particular, the "Associated Press Stylebook" (AP) and "The New York Times Manual of Style and Usage") recommend U.S. (periods, no space). The American Psychological Association (APA) "Publication Manual," which is used for writing academic papers, agrees about using the periods.

In headlines under AP style, however, it's "postal style" US (no periods). And the abbreviated form of United States of America is USA (no periods). 

Don't Use Periods—Sometimes

Scientific style guides say to omit periods in capitalized abbreviations; thus render them US and USA (no periods, no spaces). "The Chicago Manual of Style" (2017) agrees—but Chicago allows for exceptions:

"Use no periods with abbreviations that appear in full capitals, whether two letters or more and even if lowercase letters appear within the abbreviation: VP, CEO, MA, MD, PhD, UK, US, NY, IL (but see the next rule).
"In publications using traditional state abbreviations, use periods to abbreviate United States and its states and territories: U.S., N.Y., Ill. Note, however, that Chicago recommends using the two-letter postal codes (and therefore US) wherever abbreviations are used."

So what to do? Choose either U.S. or US for the piece you're writing and then stick with it, or follow the guidance that your instructor, publisher, or client prefers. As long as you're consistent in usage, neither way will look like an error.

Legal Citations in Bibliographies, Footnotes, Etc.

If you're using Chicago style and have legal-context citations in your bibliography, reference list, footnotes, or endnotes, you'll use periods, such as in Supreme Court decisions, statute numbering, and the like.

For example, when a law is incorporated into the United States Code, it has a U.S.C. designation, such as here, in this example note from Chicago: "Homeland Security Act of 2002, 6 U.S.C. § 101 (2012)." In the case of Supreme Court decisions, they're attributed to the "'United States Reports' (abbreviated U.S.)," like in this note: "Citizens United, 558 U.S. at 322." Next, a note referencing the U.S. Constitution is abbreviated "U.S. Const."

British Style Guidance

Note that British style guides recommend US (no periods, no space) in all cases: "Do not use full points in abbreviations, or spaces between initials, including those in proper names: US, mph, eg, 4am, Ibw, M&S, No 10, AN Wilson, WH Smith, etc." ("Guardian Style," 2010). "Because American and British styles differ," notes Amy Einsohn, "'CBE' ["Scientific Style and Format: The CE Manual for Authors, Editors, and Publishers"] recommends eliminating periods in most abbreviations as the most efficient way to create an international style" ("The Copyeditor's Handbook," 2007).