Let $p$ be a prime, $K$ be a finite extension of $\mathbb{Q}$ and $K_{\infty}$ be a cyclotomic $\mathbb{Z}_p$-extension of $K$ i.e. Gal$(K_{\infty}/K) \cong \mathbb{Z}_p$, the group of $p$-adic integers under addition. Then

What can we say about the prime decomposition in $K_{\infty}$ for any prime of $\mathbb{Q}$ for e.g. ramification index, order of the decomposition subgroup, finitely decomposed or not ?

The only result I know in this direction is that such extensions are unramified outside of $p$.

Can we say something more when $K=\mathbb{Q}$ ?