Practical uses of the class number formula For what number fields $K$ can we actually compute the residue of $\zeta_K(s)$ at the pole $s=1$ directly? Since the class number formula tells us that
$$\textrm{Res}_{s=1}\zeta_K(s)=\frac{2^r(2\pi)^s\textrm{reg}(K)h_K}{\# \mu(K)\sqrt{|d_K|}}$$
this doesn't seem terribly useful unless one can calculate everything except one element. The only examples that I've ever seen involve computing the residue, but I don't think that's too interesting as I would expect that the whole utility of the formula comes from computing the residue and then using whatever we know to deduce either the regulator, class number or discriminant.
The only other utility of the formula that I can come up with is if we have some relationship between the zeta-functions of some number fields. Then one could use the functional equation to compute relations between the regulators, class numbers etc. of these number fields. Picking e.g. an $S_3$ extension we find relationships between the zeta functions of the subextensions.
Could anyone elaborate on if this equation actually has any use in practical computations? Even references to papers where its used to compute something interesting would be appreciated.
 A: For any given field, you can compute the residue of the zeta function to arbitrary precision: just take the product of sufficiently many Euler factors. In practice, this can be quite slow for high degree fields, since you might need lots of factors to get the desired precision.
Of course, computations can never prove the class number 1 problem, since a computation can only check finitely many fields.
The main interest of the formula is that it provides a prototypical example of a "local-global" principle: the left hand side is something that you compute locally at each prime, while the right hand side is something manifestly global. As you probably know, the class number formula has served as a blue print for very deep and far reaching generalisations, such as the Birch and Swinnerton-Dyer conjecture for example. Most of these generalisations are wide open.
Since you also asked for references, e.g. about extracting information about fields from relations between zeta functions, here is a paper of mine in which concrete information about the Galois module structure of the units is extracted from the relationship you mention.
Edit: I failed to mention perhaps the most widely used practical application of the class number formula: when you ask MAGMA or most other algebraic number theory computer packages to compute the class group of a number field, the routines will make use of the class number formula. The way this works is as follows: it is easy to find generators of the class group (e.g. using the Minkowski bound). You then want to look for all possible relations, but how do you know when you are done? Of course, you can prove by extremely brute force that you have found all the relations, but the way the computer will do this, which is much faster, is to also find a finite index subgroup of the unit group, and as you are finding more relations for the ideal classes, you also saturate your finite index subgroup of the units, i.e. keep checking whether certain elements of your subgroup are divisible by an integer inside the unit group, thus reducing the index of your subgroup inside the full unit group. More relations lead to a smaller candidate for the class group, and further saturation leads to a smaller regulator. In the meantime, you compute the residue of the zeta function to some precision, thus knowing in advance what the product of class number and regulator should be. When you reach the expected product, you stop. Thus, most routines for computing the class group secretly also compute the unit group of the number field at the same time, and the analytic class number formula provides a stopping condition for this calculation.
