Flatness does not force isomorphic or smooth fibres

Flatness is not the same as "all fibres isomorphic", and it does not force smoothness. The standard witness over $\mathbb{Q}$ is the family $f:\mathrm{Spec}\frac{\mathbb{Q}[t,x]}{(x^2-t)}⟶\mathrm{Spec}\mathbb{Q}[t]={\mathbb{A}}_{\mathbb{Q}}^1,B=\frac{\mathbb{Q}[t,x]}{(x^2-t)}\mathrm{.}$

The equation $x^2=t$ is monic in $x$ of degree $2$, so by monic division every class in $B$ has a unique representative $a(t)+b(t)x$ with $a,b\in \mathbb{Q}[t]$; hence $B$ is free with basis $1,x$ over $A=\mathbb{Q}[t]$, therefore flat over $A$ (Under the stated choice boundary, free modules are projective and hence flat), so $f$ is a flat morphism (Flat morphism of schemes).

The fibres are computed from the tensor product (Scheme-theoretic fibre, Geometric fibres and geometric points). At $t=0$, $B{\otimes }_{A}A\mathrm{/}(t)\cong \frac{\mathbb{Q}[x]}{(x^2)}$ in which $x$ is a nonzero nilpotent: it is nonzero because $x=x^{2}q$ in $\mathbb{Q}[x]$ would give $1=xq(x)$ by cancellation in the domain $\mathbb{Q}[x]$, contradicting degrees. At $t=1$, $B{\otimes }_{A}A\mathrm{/}(t-1)\cong \frac{\mathbb{Q}[x]}{(x^2-1)}\cong \mathbb{Q}\times \mathbb{Q},$ the isomorphism being evaluation $g\mapsto (g(1),g(-1))$ with inverse $(a,b)\mapsto \frac{a+b}{2}+\frac{a-b}{2}x$ and kernel generated by the comaximal factorisation $x^2-1=(x-1)(x+1)$ (Chinese remainder theorem for pairwise comaximal ideals).

So the fibre over $t=0$ has one point (the local ring $\mathbb{Q}[x]\mathrm{/}(x^2)$ has the single prime $(x)$, Prime ideals and maximal ideals in a commutative ring), while the fibre over $t=1$ has two points. The fibres are therefore not isomorphic as schemes, and the special fibre is not even reduced. Since a morphism that is smooth at a point has geometrically regular, hence reduced, fibre there (Smooth morphism of schemes), $f$ is flat but not smooth at the origin; the visual phenomenon is the collision of the two branches $x=\pm \sqrt{t}$ as $t\to 0$, which flatness permits because flatness constrains only the $A$-module structure, not the geometry of the special fibre. No fibre-dimension theorem and no choice principle are used.