US Edition is essentially the same as the last Singapore edition before they revised their standards, but with US units and spelling. Standards is based on the California curriculum frameworks of 2007. Common Core is based on the Common Core standards (obviously!). "Based on," in the latter two cases, means that they've aligned the presentation of topics with the grade levels indicated in the respective curriculum standards.
This page from singaporemath.com has more:
http://www.singaporemath.com/FAQ_Primary_Math_s/15.htmI did not use any guides, but both I and my spouse have fairly solid math backgrounds, and I have pedagogical experience. I did purchase one of the home instructor's guides to look at, and could see how it might be valuable. It is substantially cheaper than the teacher's guide, and designed more around individual students, vs the teacher's guide's group activities.
We bought only the workbooks, which worked for us because I learned most of my math at home (I was not homeschooled), from my mother, who was educated using an Asian method, so the distance between how I learned math myself and how SM teaches it was not as great as it might have been; I was able to figure out most of the approach just from the model problems. I know other families have purchased only the textbooks, which, in retrospect, probably would have made more sense, as, in addition to the instruction, there are plenty of practice exercises in them, too. I will say that my kids liked having the workbooks to write in, though. Plus the occasional math puzzle.