For dim 2 and field $\mathbb{C}$ the calculations are elementary. First one proves by elementary group theory that the only finite subgroups of $SL(2)$ are the Cyclic, Dihedral, Tetrahedral, Octahedral and the Icosahedral groups. Now for these one can by hand write down the equations for the quotients. Look at the nice article by Slodowy titled Platonic solids, Kelinian singularities and Lie groups.

For dim 2 and field $\mathbb{C}$ the calculations are elementary. First one proves by elementary group theory that the only finite subgroups of $SL(2)$ are the Cyclic, Dihedral, Tetrahedral, Octahedral and the Icosahedral groups. Now for these one can by hand write down the equations for the quotients. Look at the nice article by Slodowy titled Platonic solids, Kelinian singularities and Lie groups. Also isolated rational singularities on surfaces were classified by Artin and include the above. Look at the paper of Artin "Isolated rational singularities on surfaces".