Answer
$S$ is linearly independent.
Work Step by Step
Assume the combination
$$a(0,0,0,1)+b(0,0,1,1)+c(0,1,1,1)+d(1,1,1,1)=(0,0,0,0).$$
We have the system
\begin{align*}
d&=0\\
c+d&=0\\
b+c+d&=0\\
a+b+c+d&=0
\end{align*}
one can see that the above system has unique solution, that is, $$a=0, \quad b=0, \quad c=0, \quad d=0.$$
Consequently, $S$ is linearly independent.