1. We need to test the truth of the two statements in probability.

A basic fact is:

• An event can have probability $0$ without being the empty set.
• An event can have probability $1$ without being the whole sample space.

This happens especially in continuous sample spaces.

2. Check statement (S1): If $P(A)=0$, then $A=\phi$.

This is false.

Counterexample

Take the experiment of choosing a real number uniformly from the interval $[0,1]$. Then $\Omega=[0,1].$ Let $A=\left\{\tfrac12\right\}.$ Clearly, $A\neq \phi$ since it contains one point. But in a continuous uniform distribution, $P\left(\left\{\tfrac12\right\}\right)=0.$ So $P(A)=0$ but $A\neq \phi$.

Hence, (S1) is false.

3. Check statement (S2): If $P(A)=1$, then $A=\Omega$.

This is also false.

Counterexample

Again take $\Omega=[0,1]$ with uniform probability, and let $A=[0,1]\setminus \left\{\tfrac12\right\}.$ Then $A\neq \Omega$ because the point $\tfrac12$ is missing. But $P(A)=1-P\left(\left\{\tfrac12\right\}\right)=1-0=1.$ So $P(A)=1$ but $A\neq \Omega$.

Hence, (S2) is false.

4. Therefore:

• (S1) is false
• (S2) is false

So the correct option is $\boxed{\text{B: both (S1) and (S2) are false}}.$