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://www.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.