There is a proper map $f: X \rightarrow Y$ of projective varieties. The Decomposition Theorem of Beilinson–Bernstein–Deligne-Gabber states that

$$Rf∗IC_X \cong \oplus_a IC_{\bar{Y_a}}(L_a)[shifts]$$

My question is: what are some interesting classes of examples maps $f: X \rightarrow Y$ such that the local systems $L_a$ we get on the right hand side above is nontrivial?

Right now I could only think of examples like finite branched coverings of projective spaces, $f: \mathbb{P}^1 \rightarrow \mathbb{P}^1$ such that $z$ gets mapped to $z^n$ for some positive integer $n$, Springer resolution, etc.