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 $p$-adic 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.(1-MN-SM)
Kronecker-Weber plus epsilon. (English summary)
Duke Math. J. 114 (2002), no. 3, 439--475.
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 Fontaine-Mazur 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, 612--637.
MR2412044 (2009m:11092) Çiperiani, Mirela; Wiles, Andrew Solvable points on genus one curves. Duke Math. J. 142 (2008), no. 3, 381--464.
The point is that it is one thing to show that two mathematical objects are isomorphic; it is another (stronger) thing to give a particular isomorphism between them. A rather concrete instance of this is in combinatorics, where if $(A_n)$ and $(B_n)$ are two families of finite sets, one could show that $\# A_n = \# B_n$ by finding formulas for both sides and showing they are equal, but it is preferred to find an actual family of bijections $f_n: A_n \rightarrow B_n$.
This is not just a matter of fastidiousness or a general belief that constructive proofs are better. When considering functorialities between various isomorphic objects, the choice of isomorphism matters. For instance, often one wants to put various isomorphic objects into a diagram and know that the diagram commutes: this of course depends on the choice of isomorphism.
In the case of class field theory, these functorialities take the form of maps between the abelianized Galois groups / norm cokernel groups / idele class groups of different fields. The isomorphisms of class field theory can be shown to be the unique ones which satisfy various functoriality properties (and some "normalizations" involving Frobenius elements), and this uniqueness is often just as useful in the applications of CFT as the existence statements.
All of this, by the way, is explained quite explicitly in Milne's (excellent) notes: you just have to read a bit further. See for instance Theorem 1.1 on page 20: "There exists a unique homomorphism...with the following properties [involving Frobenius automorphisms and functoriality]..."
As a final remark: it is important to note that the word "canonical" in mathematics does not have a canonical meaning. To say that two objects are canonically isomorphic requires further explanation (as e.g. in the Theorem I mentioned above). Even the "unique isomorphisms" that one gets from universal mapping properties are not unique full-stop [generally!]; they are the unique isomorphisms satisfying some particular property.
Best Answer
In his preface to "Rapport sur la Cohomologie des Groups", Serge Lang says that those notes "provided missing chapters to the Artin-Tate notes on class field theory". It is available in english translation under the title "Topics in the cohomology of groups".
Edit: So perhaps Lang writing that it providing missing chapters does not mean that it provides ALL missing chapters.