It seems to me that there are irrational surfaces over mathbbQ that are mathbbQv-rational for all v. (I couldn't find them in the literature, but didn't look very hard. Almost certainly they are to be found there, in papers by either Iskovskikh or Colliot-Thelene.)
Take the affine surface S given by y2+byz+cz2=f(x), where f is an irreducible cubic and b2−4c equals the discriminant D(f) of f, up to a square in mathbbQ∗, and D(f) is not a square. According to Beauville-Colliot--Thelene--Sansuc--Swinnerton-Dyer S is not mathbbQ-rational, but is stably rational. (Irrationality is Iskovskikh I think, in fact.) Via projection to the x-line a projective model V of S is a conic bundle over mathbbP1 with 4 singular fibers (one is at infinity). There is an embedding of V into a weighted projective space mathbbP(2,2,1,1); the defining equation is Y2+bYZ+cZ2=F(X,T)T, where F is the homogeneous version of f. By construction the Galois action on the 8 lines that comprise the singular fibers is via the symmetric group S3: the two lines in the fiber at infinity are conjugated, and the other six are permuted transitively.
Claim: Assume that D(f) is square-free and prime to 6. Then S is mathbbQv-rational for all v.
Proof: Suppose that the decomposition group Gv at v is cyclic. Whatever its order (1,2 or 3) there are at least 2 disjoint lines among the 8 that are Gv-conjugate, so they can be blown down to give a conic bundle over mathbbP1 with at most 2 singular fibres and a mathbbQv-point; it is well known that such a surface is mathbbQv-rational.
Now suppose that Gv=S3. Then v is non-archimedean and V has bad reduction there. In fact, exactly two of the singular fibers are equal modulo v; it follows that Gv=S3 is impossible, and we are done.
E.g., f=x3+x+1, of discriminant −31, c=8, b=1.
(This doesn't use stable rationality, but rather the fact that these surfaces, although irrational, are very close to being rational, in the sense that the action of GalmathbbQ on the lines is as small as possible subject to the surface being irrational, and the action of the decomposition groups is even smaller.)
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