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 $t = \infty$.)