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 $t = \infty$.)

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.  (See postscript. This uses characteristic zero; in small 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 $t = \infty$.)

P.S. There are other ways to prove $s>2$; for example it follows from Szpiro's inequality, which has an elementary proof via the Mason-Stothers theorem (polynomial ABC). See MO 190530, Are there nonisotrivial elliptic curves over $\mathbb{G}_m$?, for this and some related ideas. (That question is related because ${\bf G}_m({\bf C}) = {\bf CP}^1 - \{0, \infty\}$ and if an elliptic surface $\pi: X \to {\bf P}^1$ has $s \leq 2$ then one can choose the coordinate on ${\bf P}^1$ so that each bad fiber maps to $0$ or $\infty$.)