Classification of $\operatorname{Rep}D(H)$ Question
Let $H$ be a finite dimensional complex Hopf algebra and $D(H)$ its quantum double. Can we classify the simple objects in $\operatorname{Rep}D(H)$ if the representations of $H$ are well-understood?
My impression is that this is not yet completely developed. However, the more examples the better! Would you mind sharing papers/resources that deal with the representations of quantum doubles, even those that deal with specific examples?
Example: finite group algebra
For example, if $H$ is the complex group algebra of some finite group $G$, both $\operatorname{Rep}(G)$ and $\operatorname{Rep}D(G)$ are well-enough understood (to my standard). See my previous questions about this


*

*Representations of $D(G)$ as an object in the center of $\operatorname{Rep}(G)$

*Classification of $\operatorname{Rep} D(G)$
Interestingly, irreducible representations of $D(G)$ do not restrict to irreducible ones as $G$-reps. Viewing them as objects of $Z(\operatorname{Rep}G)$ reveals hidden structures among irreps of $G$: nontrivial half-braidings arise naturally!
Example: Taft algebra
As another example, the representations of the double of Taft algebras are examined here by Chen.
 A: To any skew-pairing $\lambda:U\otimes H\rightarrow k$, one can associate a hopf algebra $D(U, H)$ (built on $U\otimes H$) which is called the generalized quantum double of $U$ and $H$.
(If $H$ is finite dimensional, $U=H^{*cop}$ and $\lambda$ is the usual evaluation map, then, this corresponds to the usual quantum double $D(H)$ introduced by Drinfeld). 
In the article On the irreducible representations of generalized quantum doubles, the authors describe the irreducible representations of generalized quantum double hopf algebras $D_{\lambda_f}(U,H)$, where $\lambda_f:U\otimes H\rightarrow k$ is a skew-pairing induced by a surjective hopf algebra map $f:U\rightarrow H^{*cop}$.  
One of the main results of the paper (as i understand it) is theorem 1.1 (p.2), which describes the simple objects of $\operatorname{Rep}\big(D_{\lambda_f}(U,H)\big)$, in a way which generalizes the corresponding description of the simple objects of $\operatorname{Rep}D(G)$ (discussed in the questions linked to the OP). Then the paper proceeds in a classification of the generalized quantum double, simple modules, for the case where both $U,H$ are semisimple hopf algebras and there is a surjective map $U\rightarrow H^{*cop}$. 
Furthermore:
In The Representations of Quantum Double of Dihedral Groups, the finite dimensional,  indecomposable, left $D(kD_n)$-modules are classified, where $D_n$ is the dihedral group of order $2n$ and $k$ is an algebraically closed field, of odd characteristic $p\ |\ 2n$. 
