Is/should there be a theory of finite solvable extensions over a given base field? Could it be based on/use class field theory? Assume the base field isn't a local field.

As FC says, since solvable extensions are built up out of abelian extensions, class field theory is certainly relevant and helpful in understanding the structure of solvable extensions. On the other hand, to do this in a systematic way requires understanding class field theory of each number field in a tower "all at once". The picture that one gets in this way seems quite blurry compared to the classical goal of class field theory: to describe and parameterize the finite abelian extensions L of a field K in terms of data constructed from K itself. In the case of a number field, this description is in terms of groups of (generalized) ideal classes, or alternately in terms of quotients of the idele class group. I'm pretty sure there's no description like this for solvable extensions of any number field. What I can offer is a bunch of remarks: 1) Sometimes one has a good understanding of the entire absolute Galois group of a field K, in which case one gets a good understanding of its maximal (pro)solvable quotient. Of course this happens if the absolute Galois group is abelian. 2) Despite the OP's desire to exclude local fields, this is one of the success stories: the full absolute Galois group of a local field (complete, discretely valued field with finite residue field) is a topologically finitely presented prosolvable group with explicitly known generators and relations. 3) On the other hand, we seem very far away from an explicit description of the maximal solvable extension of Q. For instance, in the paper MR1924570 (2003h:11135) Anderson, Greg W.(1MNSM) KroneckerWeber plus epsilon. (English summary) Duke Math. J. 114 (2002), no. 3, 439475. the author determines the Galois group of the extension of Q^{ab} which is obtained by taking the compositum of all quadratic extensions K/Q^{ab} such that K/Q is Galois. Last week I heard a talk by Amanda Beeson of Williams College, who is working hard to extend Anderson's result to imaginary quadratic fields. 4) This question seems to be mostly orthogonal to the "standard" conjectural generalizations of class field theory, namely the Langlands Conjectures, which concern finite dimensional complex representations of the absolute Galois group. 5) A lot of people are interested in points on algebraic varieties over the maximal solvable extension Q^{solv} of Q. The field arithmeticians in particular have a folklore conjecture that Q^{solv} is Pseudo Algebraically Closed (PAC), which means that every absolutely irreducible variety over that field has a rational point. This would have applications to things like the Inverse Galois Problem and the FontaineMazur Conjecture (if that is still open!). Whether an explicit description of Q^{solv}/Q would be so helpful in these endeavors seems debatable. I have a paper on abelian points on algebraic varieties, in which the input from classfield theory is minimal. The two papers on solvable points that I know of (and very much admire) are: MR2057289 (2005f:14044) Pál, Ambrus Solvable points on projective algebraic curves. Canad. J. Math. 56 (2004), no. 3, 612637. MR2412044 (2009m:11092) Çiperiani, Mirela; Wiles, Andrew Solvable points on genus one curves. Duke Math. J. 142 (2008), no. 3, 381464. 

