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 $[13/31,14/31[$.