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