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. My question is

How many primes of $K_{\infty}$ lie above any prime of $\mathbb{Q}$ ?

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

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

Any references are welcome.