  An (improper) graph colouring has "defect" $d$ if each monochromatic subgraph
has maximum degree at most $d$, and has "clustering" $c$ if each monochromatic
component has at most $c$ vertices. This paper studies defective and clustered
list-colourings for graphs with given maximum average degree. We prove that
every graph with maximum average degree less than $\frac{2d+2}{d+2} k$ is
$k$-choosable with defect $d$. This improves upon a similar result by Havet and
Sereni [J. Graph Theory, 2006]. For clustered choosability of graphs with
maximum average degree $m$, no $(1-\epsilon)m$ bound on the number of colours
was previously known. The above result with $d=1$ solves this problem. It
implies that every graph with maximum average degree $m$ is
$\lfloor{\frac{3}{4}m+1}\rfloor$-choosable with clustering 2. This extends a
result of Kopreski and Yu [Discrete Math., 2017] to the setting of
choosability. We then prove two results about clustered choosability that
explore the trade-off between the number of colours and the clustering. In
particular, we prove that every graph with maximum average degree $m$ is
$\lfloor{\frac{7}{10}m+1}\rfloor$-choosable with clustering $9$, and is
$\lfloor{\frac{2}{3}m+1}\rfloor$-choosable with clustering $O(m)$. As an
example, the later result implies that every biplanar graph is 8-choosable with
bounded clustering. This is the best known result for the clustered version of
the earth-moon problem. The results extend to the setting where we only
consider the maximum average degree of subgraphs with at least some number of
vertices. Several applications are presented.
