Ramification in a Zp-extension

At least one finite prime ramifies in a Zp-extension; otherwise the entire tower would lie in a finite Hilbert class field. Ramification can occur only over $p$: local class field theory makes inertia away from $p$ an image of a unit group with finite maximal pro-$p$ quotient, whereas ${\mathbb{Z}}_p$ has no nontrivial finite subgroup. Nonzero inertia is open in ${\mathbb{Z}}_p$, so it becomes total after passing to a sufficiently high finite layer.