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 $b$): $[0/1,1/31[$, $[11/29,8/21[$, $[13/16,22/27[$, $[4/19,3/14[$, $[20/29,9/13[$, $[9/16,13/23[$, $[19/20,20/21[$, $[11/24,6/13[$, $[8/29,5/18[$, $[1/8,4/31[$, $[16/21,13/17[$, $[28/31,19/21[$, $[16/25,9/14[$, $[13/25,12/23[$, $[7/22,8/25[$, $[5/29,4/23[$, $[1/12,2/23[$, $[17/20,23/27[$, $[5/12,13/31[$, $[3/5,17/28[$, $[21/29,8/11[$, $[30/31,1/1[$, $[7/29,1/4[$, $[1/24,1/23[$, $[10/29,9/26[$, $[15/31,1/2[$, $[24/31,7/9[$, $[27/31,8/9[$, $[19/29,2/3[$, $[4/29,5/31[$, $[17/30,18/31[$, and $[13/31,14/31[$.$$\begin{array}{rl|rl|rl|rl} [0/1, & 1/31[ & [11/29, & 8/21[ & [13/16, & 22/27[ & [4/19, & 3/14[ \\ [20/29, & 9/13[ & [9/16, & 13/23[ & [19/20, & 20/21[ & [11/24, & 6/13[ \\ [8/29, & 5/18[ & [1/8, & 4/31[ & [16/21, & 13/17[ & [28/31, & 19/21[ \\ [16/25, & 9/14[ & [13/25, & 12/23[ & [7/22, & 8/25[ & [5/29, & 4/23[ \\ [1/12, & 2/23[ & [17/20, & 23/27[ & [5/12, & 13/31[ & [3/5, & 17/28[ \\ [21/29, & 8/11[ & [30/31, & 1/1[ & [7/29, & 1/4[ & [1/24, & 1/23[ \\ [10/29, & 9/26[ & [15/31, & 1/2[ & [24/31, & 7/9[ & [27/31, & 8/9[ \\ [19/29, & 2/3[ & [4/29, & 5/31[ & [17/30, & 18/31[ & [13/31, & 14/31[. \end{array}$$
