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Some added observations on what is likely to be a very complicated problem:

1) The 2010 Selecta Math. paper by Fresse and Melnikov which you mention was posted here, but the published version is somewhat longer. I haven't checked the precise differences, but this is always a hazard when consulting arXiv preprints. The joint work of Melnikov with various people has uncovered many of the known results about singularities of irreducible components of Springer fibers. This builds on work of Francis Fung in type $A$, for example, and mostly deals with aspects that can be formulated combinatorially.

2) As far as I can see, the geometry of Springer fibers is little undeerstood, beyond the older work of Spaltenstein which determined in general the dimensions of the fibers and proved their the equidimensionality of their components. So the refined study of singularities (or normality) is likely to require some new ideas.

3) While I agree with what Alexander Woo says in his first paragraph, I don't agree with his reading of the 1979 Kazhdan-Lusztig paper. As their introduction indicateds, they were motivated by several different-looking problems including Springer's representations of Weyl groups (on cohomology of fibers in the Springer desingularization of the unipotent variety), the indirectly related problem of working out singularities of Schubert varieties, and the even more indirectly related problem of understanding algebraic work on Verma modules and primitive ideals in enveloping algebras. While those subjects do have connections, they are rather subtle, as Geordie's comment indicates.

4) One aspect which remains mysterious to me is the rough analogy between two pictures: Schubert varieties in the flag variety and their partial ordering (corresponding to the Chevalley-Bruhat ordering in the Weyl group); unipotent classes or nilpotent orbits and their closure ordering. In each case you have varieties which are sometimes not smooth, and might or might not even be normal. Moreover, the two pictures are connected indirectly by the fact that the Springer fibers live in the cotangent bundle of the flag variety. All of this requires unpacking, even in type $A$ where the combinatorics is better-behaved.

5) Concerning Springer fibers and their irreducible components, the cohomology of a fiber turns out to be important in representation theory of different flavors. On the other hand, Slodowy's work on slices to nilpotent orbits leads to smooth varieties in the cotangent bundle which have twice the dimension of the corresponding Springer fiber but lead to the same equivariant K-theory in Lusztig's further work. He has studied these equivariant cohomology groups and their canonical bases extensively. In some ways this may be a more fruitful aspect of Springer fibers than the nonsingularity of their components, but there are many unknowns.

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Some added observations on what is likely to be a very complicated problem:

1) The 2010 Selecta Math. paper by Fresse and Melnikov which you mention was posted here, but the published version is somewhat longer. I haven't checked the precise differences, but this is always a hazard when consulting arXiv preprints. The joint work of Melnikov with various people has uncovered many of the known results about singularities of irreducible components of Springer fibers. This builds on work of Francis Fung in type $A$, for example, and mostly deals with aspects that can be formulated combinatorially.

2) As far as I can see, the geometry of Springer fibers is little undeerstood, beyond the older work of Spaltenstein which determined in general the dimensions of the fibers and proved their equidimensionality. So the refined study of singularities (or normality) is likely to require some new ideas.

3) While I agree with what Alexander Woo says in his first paragraph, I don't agree with his reading of the 1979 Kazhdan-Lusztig paper. As their introduction indicateds, they were motivated by several different-looking problems including Springer's representations of Weyl groups (on cohomology of fibers in the Springer desingularization of the unipotent variety), the indirectly related problem of working out singularities of Schubert varieties, and the even more indirectly related problem of understanding algebraic work on Verma modules and primitive ideals in enveloping algebras. While those subjects do have connections, they are rather subtle, as Geordie's comment indicates.

4) One aspect which remains mysterious to me is the rough analogy between two pictures: Schubert varieties in the flag variety and their partial ordering (corresponding to the Chevalley-Bruhat ordering in the Weyl group); unipotent classes or nilpotent orbits and their closure ordering. In each case you have varieties which are sometimes not smooth, and might or might not even be normal. Moreover, the two pictures are connected indirectly by the fact that the Springer fibers live in the cotangent bundle of the flag variety. All of this requires unpacking, even in type $A$ where the combinatorics is better-behaved.

5) Concerning Springer fibers and their irreducible components, the cohomology of a fiber turns out to be important in representation theory of different flavors. On the other hand, Slodowy's work on slices to nilpotent orbits leads to smooth varieties in the cotangent bundle which have twice the dimension of the corresponding Springer fiber but lead to the same equivariant K-theory in Lusztig's further work. He has studied these equivariant cohomology groups and their canonical bases extensively. In some ways this may be a more fruitful aspect of Springer fibers than the nonsingularity of their components, but there are many unknowns.