References for $K_{4k}(\mathbb{Z})$

$K_8(\mathbb{Z})=0$

A proof of the result announced by Elbaz-Vincent and mentioned above can now be found in:

• Mathieu Dutour-Sikirić, Philippe Elbaz-Vincent, Alexander Kupers, and Jacques Martinet, Voronoi complexes in higher dimensions, cohomology of $GL_N(\mathbb{Z})$ for $N\geq8$ and the triviality of $K_8(\mathbb{Z})$, arXiv:1910.11598v1 (submitted on October 25, 2019)

A shorter proof, which (unlike the one above) does not require additional computer calculations, is presented in:

• Alexander Kupers, A short proof that $K_8(\mathbb{Z})=0$, http://people.math.harvard.edu/~kupers/files/k8zshorter.pdf (version of October 21, 2019)

The very last remark in Kupers’s short note addresses the case of $K_{12}(\mathbb{Z})$:

Remark 2.6. If Conjecture 2 of [CFP14] were true with coefficients in $\mathbb{Z}[1/((n+1)!)]$ instead of $\mathbb{Q}$, it could be used to prove that $K_{12}(\mathbb{Z})=0$ in a similar manner.

[CFP14] Thomas Church, Benson Farb, and Andrew Putman, A stability conjecture for the unstable cohomology of $SL_n(\mathbb{Z})$, mapping class groups, and $Aut(F_n)$, Algebraic topology: applications and new directions, 55–70, Contemp. Math., vol. 620, Amer. Math. Soc., 2014, MR3290086

Well, the consensus seems to be that this is an open problem, for $k >1$.

This is a quote from A. Raghuram's paper on the volume "The Bloch–Kato Conjecture for the Riemann Zeta Function" (page 8), published in april 2015.

[...] it is expected $K_{4a}(\mathbb{Z})=0$. This is proven for $a=1$ and is open as yet for $a\geq 2$.

Given that other contributors include Stephen Lichtenbaum and Manfred Kolster, this most likely represents the current knowledge at least at the time of the conference (2012).

If someone has any update (particularly regarding Philippe Elbaz-Vincent's claim, see the comments above), feel free to answer or comment.

As you probably know, this is related to the Vandiver conjecture. So the references at http://en.wikipedia.org/wiki/Kummer%E2%80%93Vandiver_conjecture might help.