Noether's Problem was introduced by Emmy Noether in [4]:
Let $\mathsf{k}$ be a field and $K=\mathsf{k}(x_1,\ldots,x_n)$ be a purely transcendental extension. Let $G<S_n$ be a group acting by permutations on the set of variables, in a transitive fashion. $G$ induces a group of algebra automorphisms of $K$, which for simplicity again will be called $G$. Noether's Problem asks: for what finite permutation actions is $K^G$ a purely transcendental extension of $\mathsf{k}$?
Noether introduced the problem due to its connection to the Inverse Galois Problem (see [2]). Many interesting cases of group actions giving positive solution to this problem were discorvered (see [1]).
Eventually, over $\mathbb{Q}$, counter-examples to Noether's Problem were discovered simultaneously by Swan and Vokresenskii ([6],[7]), and then Lenstra (again over $\mathbb{Q})$ [3] classified all abelian groups for which Noether's Problem fails (and discovered that the smallest group that can give a counter-example to Noether's Problem is $C_8$).
Finally, Saltman in [5] gave counter-examples in algebraically closed fields. Being more precise, he constructed an infinite family of $p$-groups that give counter-examples for all algebraically closed fields whose characteristic does not divides $|G|$.
Much more counter-examples were found after that: a recent survey discussing this is [1].
The literature on counter-examples to Noether's Problem is vast, but in all cases I know of, the counter-examples shows not only that $K^G$ is not rational; it is also not even stably-rational.
I am very interested in a possible counter-example to Noether's Problem such that $K^G$ is not rational but it is stably-rational. Any known reference on this would be greatly appreciated; I will also accept as answers 'folkore' examples.
References:
[1] A. Hoshi, "Noether's problem and rationality problem for multiplicative invariant fields: a survey", arXiv:2010.01517.
[2] C. U. Jensen, A. Ledet, N. Yui "Generic Polynomials: Constructive Aspects of the Inverse Galois Problem"
[3] H.W. Lenstra Jr., "Rational functions invariant under a finite abelian group"
[4] E. Noether, "Gleichungen mit vorgeschriebener Gruppe"
[5] D.J. Saltman, "Noether's problem over an algebraically closed field"
[6] R.G. Swan, "Invariant rational functions and a problem of Steenrod"
[7] V.E. Voskresenskii, "Algebraic tori"
Bonus historical question: in older literature on the subject, some author's call Noether's Problem 'Noether's Conjecture', which seems to imply that she belived that her problem would have a positive solution for every group. Unfortunately, I can't read German, and hence I don't know what was the opinion she expressed when she introduced the problem. Does anyone knows if she in fact believed that her problem would have a positive solution for all groups?
