I have heard that Progress towards the Inverse Galois problem over $\mathbb{Q}$
is very well documented for sporadic groups ($M\_{23}$ is the only case open) and for $PSL\_n(q)$.


From where I can read more about these results in details. Like papers, articles, books, etc.



On progress towards inverse Galois problem over rationals
All the sporadic groups except for $M\_{23}$ and $M\_{24}$ are realized over $\mathbb{Q}$ using the rigidity criterion. This technique is explained in all three main textbooks on inverse Galois theory:


* Jean-Pierre Serre - Topics in Galois theory
* Helmut Volklein - Groups as Galois Groups
* Gunter Malle & B. H. Matzat - Inverse Galois theory


For example Volklein explains some of the details applied to the case of $M\_{12}$ and the Monster group $M$, while Malle & Matzat present simplified proofs of much more sporadic groups. In the case of the the Monster, you might want to look at Thompson's classic paper:


* John G. Thompson - "Some finite groups which appear as $\mathrm{Gal} (L/K)$, where $K ⊆ \mathbb{Q}(μ\_n)$" (1984)


There's also this very informative answer [here at MO](https://mathoverflow.net/q/13861) by user Wanderer.


The case of $M\_{24}$, which escapes rigidity, is realized in:


* B. H. Matzat - "Rationality Criteria for Galois extensions" (1987)


The progress on $\mathrm{PSL}\_n(q)$ is less uniform. Volklein in his book covers his own regularity result in the case of $n$ even and $n \geq q$, using weak rigidity and braiding action. He also looks at the $\mathrm{PSL}\_2(q)$ case, for which Serre (chapter 5) is also a good reference.


There's of course plenty of other results and special cases known. Malle-Matzat is by far the more comprehensive and includes most of the references that you might need.


