Let $K_n = \Bbb Q(\mu_{p^{n+1}})$ and let $A_n$ be it's class group. Iwasawa theory tells us a lot about the $p$-part of $A_n$. For instance, we know quite a lot about how it varies with $n$.

I am interested in the $\ell$-part of $A_n$ where $\ell \neq p$ is a prime. What is known about these groups? For instance, do we know which primes can occur in the class numbers of these extensions? Do we know the order of growth of the prime powers that do occur?

Since none of the standard sources seem to cover these questions (despite being quite natural), I suspect this is quite a difficult question. What is the main difficulty in extending the methods of classical Iwasawa theory to this case?

For instance, going through the proof of the p-part of the class number in a $\Bbb Z_p$ extension (Chapter 13, Sections 1-3 of Washington's Cyclotomic fields), we have a problem while classifying modules over the corresponding Iwasawa algebra $\Bbb Z_\ell[[\Bbb Z_p]]$. If $\ell = p$, then this ring would be isomorphic to $\Bbb Z_p[[t]]$ and one can classify modules over this field (upto finite kernel and cokernel).

What other problems arise like this?