Good question--and unfortunately there's not one simple answer. In general, newspaper style guides (The Associated Press Stylebook and The New York Times Manual of Style and Usage) recommend "U.S." (periods but no space).
However, scientific style guides say to omit periods in capitalized abbreviations, thus "US" (no periods, no space).
The Chicago Manual of Style (2010) agrees--but allows for exceptions:
3. 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 rule 4).My advice? Choose one style or the other, and then stick with it consistently.
4. 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 . . ..