i) Find subsets $E ⊆ S _1 ⊆ S _2 ⊆ S _3 ⊆ \Bbb Q$ such that $E$ has a least upper bound in $S_ 1$ , but does not have any least upper bound in $S_ 2$ , yet does have a least upper bound in $S _3$.
ii) Can there exist an example with the properties asked for in i) such that $E = S_ 1$ ?.
I think that I have answer for ii): No, $E=S_1$ is not possible, because if $E=S_1$, then since $E$ has a least upper bound in $S_ 1$, say, $s=\sup E$, we see that $s\in S_1\subset S_2$. So, $E$ has a least upper bound in $S_2$, contrary to our requirement.
I can't find answer to i)
Your ii) is ok.
As (i): Let $E=\{\,x\in\Bbb Q\mid x<0\,\}$, $S_1=E\cup\{42\}$, $S_2=\Bbb Q\setminus \{0\}$, $S_3=\Bbb Q$.