Evidence for $Q^{\operatorname{solv}}$ being pseudo-algebraically-closed

So far as I know, there is no compelling evidence to support this conjecture. (And some leading arithmetic geometers think it is false.) Rather, there are some very interesting consequences of this conjecture, e.g. a solution of the Inverse Galois Problem over Q^{solv}: in other words, for any finite group G, there exists a tower of radical extensions

K_0 = Q, K_1 = K_0(a_0^{1/n_0}) < K_2 = K_1(a_1^{1/n_1}) < ... <= K_n

and a Galois extension L/K_n with Galois group isomorphic to G. It also shows that geometrically irreducible algebraic varieties "acquire rational points" in a very different way from irreducible zero-dimensional varieties (which have a unique minimal splitting field which need not be solvable).

Of course, interesting things which follow from a conjecture are, if anything, evidence against the truth of the conjecture, although they support the claim that the question is interesting.

There is one impressive result towards this conjecture, namely the Ciperiani-Wiles theorem: let C_{/Q} be a genus one curve with points everywhere locally and semistable Jacobian elliptic curve E. Then C(Q^{solv}) is nonempty.

On the negative side, there is a paper of Ambrus Pal which constructs, for each sufficiently large integer g, a curve C of genus g over a field K which does not admit any points over the maximal solvable extension of K. (Here K is not a number field.)

On the other hand, as far as I know, it is still open to find an absolutely irreducible variety V/Q which fails to have rational points over the maximal metabelian extension of Q, i.e., over (Q^{ab})^{ab}. For some thoughts about this, see

http://alpha.math.uga.edu/~pete/abeliantalk.pdf

ADDENDUM:

I forgot to address the last part of the question: what about global function fields?

As I alluded to above, there are counterexamples over the function field of a sufficiently complicated ground field, like Q. Of course you mean a finite extension of F_q(T), in which case I think absolutely nothing is known. In particular, I believe the analogue of Ciperiani-Wiles is open here, and might not be a straightforward adaptation, since C-W uses results on the modularity of elliptic curves. This could make a nice thesis problem...but I would talk to Mirela Ciperiani before doing any serious work on it.

I'll respond to this question with another question. Given a curve C/Q, there is a solvable extension K/Q such that C/K has local points everywhere. (At least I think this is is the case; it requires some kind of argument on approximating local solvable extensions by global ones, which I've seen done before.) Some people believe that the Brauer-Manin obstruction is the only obstruction to the Hasse principle for curves. If that were the case, would it say anything about your question?