We study various problems arising in higher geometry using derived Lie $\infty$-groupoids and algebroids. We construct homotopical algebras for derived Lie $\infty$-groupoids and algebroids and study their homotopy-coherent representations. Then we apply these tools in studying singular foliations and their characteristic classes. Finally, we prove an $A_{\infty}$ de Rham theorem and higher Riemann-Hilbert correspondence for foliated manifolds.