As one of the most frequent supporters of the thesis that Lilly and Becca are only half-siblings, I say:
Frank may not be Becca's father, but statistically he's still more likely than Trevor (Lilly's father).
My theory is that Shannon had a girls' night out, got drunk, hooked up with a guy and got knocked up. Voila, Becca on her way.
Her parents didn't think it was that cool, so they sent the pregnant Shannon off to her aunt's house so the neighborhood wouldn't find out that their 16 year old daughter was pregnant.
There she graduates from high school, finds a job in the burger joint and meets Trevor, with whom she then starts dating. Becca is already in the world at this point, between 2-3 years old. Trevor then gets Shannon pregnant at some point, Lilly is on her way, then marries Shannon (whether out of love or social pressure I'll leave to everyone to decide) and adopts Becca, with whom he is presumably rather distant, which she also notices and reciprocates (interestingly enough, a similarly cool situation between Frank and Lilly, except that Frank was always trying to establish a relationship here).
But I'm willing to let CNG prove me wrong if he gives us an infodrop on Shannon's past, which could at least be addressed with the question of Becca's father in her side story. Other mysteries surrounding Shannon, on the other hand, would have to be revealed somehow differently (like the why of Shannon vs Jessica), because those are secrets that would probably have to be pried from Shannon's cold, dead hands. And I doubt she keeps a diary.