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Based on the comments (in particular, when I say "isomorphism classes of ... form a set", what I really mean is "there is a set such that any ... is isomorphic to one element of this set"; or I could argue that what I say makes sense using Scott's trick : this all should be equivalent anyway), here are some examples :

1- Given a cardinal $\kappa$, a first order language $L$ and a theory $T$ in $L$, the isomorphism classes of models of $T$ form a set. An interesting special case is when you take $L$ to consist entirely of function symbols and $T$ a list of universally quantified equations; then knowing that this is a set actually helps in one of the usual proofs of the Birkhoff variety theorem.

It's important to note that I restricted to a first order language, but actually there's no need for symbols in $L$ to be finitary, as long as $L$ is a set (and so its symbols have bounded (possibly infinite) arity), this works and the proof is the same.

2- If you include "second countable" in the definition of manifold (I don't know how consensual that is, my teachers did it that way), then diffeomorphism classes of manifolds form a set. This can be interesting to know in some considerations where we have functors defined on $\mathrm{Diff}$, and so to deal with functor categories, knowing that this category is essentially small is interesting (e.g. to see that prestacks on $\mathrm{Dif}$ actually form a category, or cobordism categories)

3- Given a cardinal $\kappa$, homeomorphism classes of compact Hausdorff spaces of cardinality $\leq \kappa$ form a set. This follows because the topology on a compact Hausdorff space is completely determined by the $\lim$ function $\beta X\to X$ ($\beta X$ is the set of ultrafilters on $X$). I don't know how useful that is, but I used it (rather a trivial variation of it) in a bachelor project to prove the existence of a universal minimal flow of a topological group.

4- (part of ) 1- and 3- generalize cleanly to monadic categories : whenever a category of objects is equivalent to $\mathbf{Set}^T$, the Eilenberg-Moore category of $T$-algebras for a certain monad $T$, then for any cardinal $\kappa$ there is a set of isomorphism classes of objects of this category of cardinality $\leq \kappa$. In 1-, $T$ is the monad associated to an algebraic theory, in 4- $T$ is the ultrafilter monad.

One can modify slightly 4 (or 1) to get that isomorphism classes of "finitely generated stuff" usually form a set : $R$-modules for any ring $R$ for instance.

5- Homeomorphism classes of separable metric spaces (in fact, isometry classes of separable metric spaces) form a set. In fact, with the ultrafilter trick as above, given a cardinal $\kappa$, homeomorphism classes of Hausdorff spaces that have a dense subset of size $\leq \kappa$ form a set.

6- As Randall pointed out in the comments, given a model category $C$ and objects $X,Y\in C$ it is not clear at all that $\hom_{\mathbf{Ho}(C)}(X,Y)$ should be a set. In fact if we don't take a model category but just a category with weak equivalences and localize, in general these don't form sets. But in a model category, fibrant and cofibrant replacements and various lemmas ensure that $\hom_{\mathbf{Ho}(C)}(X,Y)$ is (isomorphic to) a quotient of a hom-set in $C$ and is therefore essentially a set.

7- An example that is not "isomorphism classes of", that is also a bit weird, but that has some interesting (philosophical) implications : the class of all sets $x$ such that $

eg\mathrm{(Fermat's\, last\, theorem)}$ is a set (known since the 90's only !)