Since you mentioned both miles and awards, I'll assume you're talking about redeeming a Star Alliance award with US.

As far as I know, but you should be verifying the specific rules with Dividend Miles, but in general if you're going on a Star award you must:

a) use more than one Star carrier. So if you book a flight pair on UA, say NYC-LHR-NYC, you'll be under the US-UA rules, rather than the Star rules: NYC-LHR/UA, LHR-FRA-NYC/LH would be a Star award

b) awards must be on the most direct routing between two city pairs you want, so for example, you shouldn't be allowed to do NYC-LAX-LHR, but you could go NYC-FRA-LHR since in transits a Star hub in an permissible routing between NYC and LHR.

c) You will find that you will be more limited by airline award availability than the "rules" about which carrier you can have. For example, you might want to try SQ for NYC-FRA, but LH may only have seats in the appriate class. Obviously if both carriers have availability then you be able to choose.

d) In general, Star awards within regions (eg: North America is a region, Europe is a region, etc) do NOT allow stopovers, however awards BETWEEN regions do, however the stopover usually needs to, again, be on the permissible routing between the points you want. And you can't have a stopover in another region (eg: you couldn't go LAX-SIN-FRA on one award).

Hope this helps as a starter....