Are there special or interesting non-principal ultrafilters? While non-principal ultrafilters cannot be explicitly constructed that doesn't mean that one cannot uniquely specify one of them under the assumption of their existence.

Is there an example of a uniquely determined non-principal ultrafilter? Say on $\Bbb N$.

If thats not possible is it common to separate non-principal ultrafilters into different types? Or are they most often treated as an unreachable homogenous blob which provides theorems?
 A: You cannot expect to be able to pinpoint any unique non-principal ultrafilter precisely because such filters cannot be described explicitly. A heuristic argument is that if some property described precisely a non-trivial ultrafilter (say on $\mathbb N$), then presumably that property is prescriptive enough to identify just one ultrafilter. But then you would expect to be able to construct the ultrafilter. Since that cannot be done, it must be the case that the property is just as vague as the axiom of choice is (or whatever choice principle is required on the way to constructing the object).
A: Non-principal ultrafilters do have types and they are studied quite a lot in set theory and model theory.
On the set theoretic side you have ultrafilters that are complete, descendingly complete, you have normal ultrafilter, weakly normal ultrafilters, and you have fine ultrafilters. We have systems of ultrafilters that are coherent in a certain technical sense called "extenders".
When they are sufficiently complete we can study their reflection properties: if $U$ is a $\kappa$-complete ultrafilter on $\kappa$, it can be that there is some $A\in U$ such that for all $\alpha\in A$, $\alpha$ also carries an $\alpha$-complete ultrafilter.
These properties are the defining properties of many large cardinal axioms, and in $\sf ZFC$ we can convert them into elementary embedding that have certain closure properties.
On the other hand, a regular ultrafilter is somehow the strong negation of a complete ultrafilter. These tend to have nice properties in applications to ultrapowers and ultraproducts. For example, if $U$ is a regular ultrafilter on $I$, and $M$ is any set, then $|M^I/U|=|M|^{|I|}$.
A: Here are two situations in which we do have a uniquely identifiable nonprincipal ultrafilter; the first of these is a bit silly (although not as silly as it may first appear), but the second is quite serious and important.

First, let's work in $\mathsf{ZF}$. An amorphous set is an infinite set which cannot be partitioned into two infinite subsets. Amorphous sets can carry some structure (e.g. it's consistent with $\mathsf{ZF}$ that there is an amorphous set which can be partitioned into pairs), but they're quite limited in this regard.
Now consider the cofinite filter on an amorphous set $A$. Since $A$ is amorphous this filter is a nonprinicpal ultrafilter - and in fact the unique nonprincipal ultrafilter on $A$. Yay!

OK, that was silly. What about a $\mathsf{ZFC}$ example?
Well, in general this isn't something we can hope for - despite the fact that there are genuine differences between nonprincipal ultrafilters on the same set (for examples on $\omega$ consider e.g. the topology and algebra of $\beta\omega$ or the Rudin-Keisler ordering). However, we can sometimes get uniqueness in a surprising way: assuming that $\mathsf{ZFC}$ + "There is a measurable cardinal" is consistent in the first place, so is the following:

$\mathsf{ZFC}$ + "There is a unique measurable cardinal" + "There is exactly one normal ultrafilter on that measurable cardinal."

In fact the situation is even better: if $M\models\mathsf{ZFC}$ and $\kappa$ is measurable in $M$, there is a minimal inner model of $M$ which thinks that $\kappa$ is measurable, and in this inner model there is exactly one normal ultrafilter on $\kappa$. This is treated in Kanamori's book (Theorem $20.10$), and is extremely surprising and cool; it's the start of the development of inner model theory for large cardinals.
