I was wondering if there are any wellknown results or hunches about whether the nontrivial zeroes of Riemannzeta (or zeta/Lfunctions in general) are algebraic or not.

There is a paper by A. E. Ingham, "On two conjectures in the theory of numbers", Amer. J. Math. 64 (1942), 313319, where he shows that if the ordinates of the nontrivial zeros of the Riemann zetafunction are linearly independent over $\mathbb{Q}$ then Merten's conjecture is false. This is, of course, weaker than the RubinsteinSarnak conjecture, but related and much earlier. 


Every nontrivial zero of every Lfunction, besides possible zeros at $s=1/2$, is conjectured to be of the form $s=1/2+i\gamma$ with $\gamma$ real (GRH) and transcendental. I learned this from (for example) the RubinsteinSarnak paper on Chebyshev biases, but they were not the first to enunciate it. 

