  In the present paper, we investigate whether an embedding of a decomposition
space $\mathcal{D}\left(\mathcal{Q},L^{p},Y\right)$ into a given Sobolev space
$W^{k,q}(\mathbb{R}^{d})$ exists. As special cases, this includes embeddings
into Sobolev spaces of (homogeneous and inhomogeneous) Besov spaces,
($\alpha$)-modulation spaces, shearlet smoothness spaces and also of a large
class of wavelet coorbit spaces, in particular of shearlet-type coorbit spaces.
  Precisely, we will show that under extremely mild assumptions on the covering
$\mathcal{Q}=\left(Q_{i}\right)_{i\in I}$, we have
$\mathcal{D}\left(\mathcal{Q},L^{p},Y\right)\hookrightarrow
W^{k,q}(\mathbb{R}^{d})$ as soon as $p\leq q$ and
$Y\hookrightarrow\ell_{u^{\left(k,p,q\right)}}^{q^{\triangledown}}\left(I\right)$
hold. Here, $q^{\triangledown}=\min\left\{ q,q'\right\} $ and the weight
$u^{\left(k,p,q\right)}$ can be easily computed, only based on the covering
$\mathcal{Q}$ and on the parameters $k,p,q$.
  Conversely, a necessary condition for existence of the embedding is that
$p\leq q$ and
$Y\cap\ell_{0}\left(I\right)\hookrightarrow\ell_{u^{\left(k,p,q\right)}}^{q}\left(I\right)$
hold, where $\ell_{0}\left(I\right)$ denotes the space of finitely supported
sequences on $I$.
  All in all, for the range $q \in (0,2]\cup\{\infty\}$, we obtain a complete
characterization of existence of the embedding in terms of readily verifiable
criteria. We can also completely characterize existence of an embedding of a
decomposition space into a BV space.
