Not sure what to say in general, but in practice one often uses a semismall resolution. For example if your sheaf $IC(L)$ is given as $\pi_* C_Z$ (pushforward of constant sheaf, or more generally orientation or dualizing sheaf if $Z$ is not smooth) for a proper $Z\to X$ you can calculate derived self-maps $$Hom(\pi_*C_Z,\pi_*C_Z)=
Hom(C_Z, \pi^!pi_*C_Z).$$ Now write $\pi_1,\pi_2$ for the two projections from $Z\times_X Z\to Z$. Then we calculate further 

$$Hom(C_Z, \pi^!pi_*C_Z)=R\Gamma_Z(\pi_{1,*}\pi_2^!C_Z)=R\Gamma_{Z\times_X Z}(\omega_{Z\times_X Z}),$$

i.e., Borel-Moore homology of $Z\times_X Z$ (the famous example of this being $X$ the nilpotent cone, $Z$ the Springer resolution, and $Z\times_X Z$ the Steinberg variety, cf. the book of Chriss-Ginzburg). This works in the equivariant setting (i.e. for sheaves on stacks) equally well, giving equivariant Borel-Moore homology.