What is the standard resolution of singularities, for the nilpotent cone (of the adjoint representation) for the symplectic group? I know how to do this for the general linear group, but am having trouble finding a good reference for the symplectic group. I understand it uses Richardson orbits. (I do know what the closure ordering should be though).
Ben gave the general answer above. If you care specifically about the symplectic group and are interested in a "flaglike" description of its flag variety, then one exists. It is given by all halfflags of isotropic subspaces (this is just like for $SL_n$, the symplectic group acts transitively and the stabilizer of the standard halfflag will be the standard Borel). With this description, it's just as straightforward computing Springer fibers and the like as it is for the $SL_n$ case, which you're presumably familiar with. A reference for these flaglike descriptions can be found in the section of Fulton and Harris on "Homogeneous Spaces" (there's a similar description for the special orthogonal groups). 


It's the Springer resolution; this works for all semisimple groups. The resolution is the moment map from the cotangent bundle of $G/B$ to $\mathfrak{g}^*$. Look at section 6 of Ginzburg's notes. Not to steal Mike's thunder, but there's an even more specific description of this resolution: given a nilpotent element $X$ of the symplectic Lie algebra, the fiber in this resolution over it is the space of complete flags $V_1\subset V_2\subset \cdots \subset \mathbb{C}^{2n}$ such that $V_i$ and $V_{2ni}$ are symplectic orthogonal (this immediately implies that all the spaces in this flag are isotropic or coisotropic) which are preserved by $X$ ($XV_i\subset V_{i1}$). 

