S701P:
Tiger Airways is run completely separately from SIA. They don't even coordinate the use of their ground handling facilities, fuel, maintenance etc in SIN.
Further, SQ only owns 49% of TR. After the IPO, it will probably fall to under 40%. As such, TR isn't even an SQ subsidiary and SQ can't exert majority control over it. Considering SQ's equity was 49% from the beginning, it was probably intentional that they didn't intend for TR to be a subsidiary. Since SQ owns 49% of Virgin Atlantic, you can probably see the kind of control SQ has over Virgin Atlantic (which is almost none)
It is very unlikely you will see Tiger operate widebodies in SIN or Australia, and back in the days where the yields from SYD to LAX were great, SIA only wanted to operate their aircraft across the Pacific. Today, SQ isn't even interested in operating that route, since VA and DL have lowered yields to such a great extent.