The minimal $s$ is $3$.
It is attained by several elliptic K3's, including $y^2 = x^3 + (t^2-t)^4$ which has IV* fibers at $t = 0, 1, \infty$ and no other singular fibers.
The comment by Ariyan Javanpeykar gives one argument that $s$ can be no smaller. (This uses characteristic zero; in positive characteristic $s$ can be as small as $1$, e.g. in characteristic 2 the elliptic K3 surface $y^2 + y = x^3 + t^9$ has only one reducible fiber, at $t = \infty$.)