The value of $s(1)$ is 31, because the maximal number of points that can be placed on the unit interval satisfying the given constraits is 32. While the computation of the exact value of $s(0)$ takes a few milliseconds (program written in C), the computation of the exact value of $s(1)$ takes about 2 days. So, it appears that knowledge of $s(2)$ will require much more computing effort.
Here goes one possible solution with 32 points ($[a,b[$ means an interval of the real line closed at $a$ and open at $b$): $$\begin{array}{rl|rl|rl|rl} [0/1, & 1/31[ & [11/29, & 8/21[ & [13/16, & 22/27[ & [4/19, & 3/14[ \\ [20/29, & 9/13[ & [9/16, & 13/23[ & [19/20, & 20/21[ & [11/24, & 6/13[ \\ [8/29, & 5/18[ & [1/8, & 4/31[ & [16/21, & 13/17[ & [28/31, & 19/21[ \\ [16/25, & 9/14[ & [13/25, & 12/23[ & [7/22, & 8/25[ & [5/29, & 4/23[ \\ [1/12, & 2/23[ & [17/20, & 23/27[ & [5/12, & 13/31[ & [3/5, & 17/28[ \\ [21/29, & 8/11[ & [30/31, & 1/1[ & [7/29, & 1/4[ & [1/24, & 1/23[ \\ [10/29, & 9/26[ & [15/31, & 1/2[ & [24/31, & 7/9[ & [27/31, & 8/9[ \\ [19/29, & 2/3[ & [4/29, & 5/31[ & [17/30, & 18/31[ & [13/31, & 14/31[. \end{array}$$
