$\newcommand{\Gal}{\mathrm{Gal}} \newcommand{\Q}{\mathbf Q} \newcommand{\Z}{\mathbf Z} \newcommand{\F}{\mathbf F}$

Abbreviate $K=\Q(\zeta_n)$. Note first that a galoisian extension $E$ of $K$ need not be galoisian over $\Q$, so I'm assuming that you are considering only those $E$ which are. We then have an exact sequenece
$$
1\to\Gal(E|K)\to\Gal(E|\Q)\to\Gal(K|\Q)\to1
$$
in which the last group is $(\Z/n\Z)^\times$, of order $\varphi(n)$. A sufficient condition for the sequence to split is : the degree $[E:K]$ is prime to $\varphi(n)$ (Schur-Zassenhaus). I don't think there is a classification of all such extensions.

Note finally that this answer does not depend on the fact that $K$ is the cyclotomic field of level $n$, or even the fact that the base field is $\Q$. It applies to any galoisian tower $E|K|F$: the associated short exact sequence
$$
1\to\Gal(E|K)\to\Gal(E|F)\to\Gal(K|F)\to1
$$
splits if the degrees $[E:K]$, $[K:F]$ are mutually prime.

**Addendum** (at Alex Bartel's suggestion): Let's return to the case $F=\Q$, $K=\Q(\zeta_n)$, $\Delta=\Gal(K|\Q)$, and suppose that $n$ is a *prime* $p$, for simplicity. Kummer theory tells us that abelian extensions $E|K$ of exponent dividing $p$ correspond bijectively to subgroups $D\subset K^\times/K^{\times p}$ under $E=K(\root p\of D)$; such an $E$ is galoisian over $\Q$ if and only if the subgroup $D$ is $\Delta$-stable. When such is the case, we get examples of the kind of extensions envisaged in the question, with "split Galois group". I guess the group $\Gal(E|\Q)$ will be commutative if and only if the $\Delta$-action on the $\F_p$-space $D$ is via the "mod $p$" cyclotomic character, namely the canonical isomorphism $\Delta\to\F_p^\times$.