Most categories we encounter in daily mathematics life ($\mathrm{Grp}$, $\mathrm{Ab}$, $\mathrm{Ring}$, $\mathrm{Top}$, $\mathrm{Set}$...) are not small.
I've been looking around here and elsewhere on the internet for a repository of examples of small categories but I've only found a few examples. So I wanted to make this big-list question with examples of small categories, preferably one example per answer (write multiple answers if you have multiple examples, preferably avoiding duplicates).
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Let $k$ be a field. The category $\mathbf{Mat}_k$ is defined as follows:
The set of objects is the set of natural numbers $\mathbf{N}_0$. Further, define $\operatorname{Hom}(n, m) = \mathrm{M}_{m \times n}(k)$ as the set of matrices. Composition is given by multiplication of matrices.
This category is equialent to the category of finite-dimensional $k$-vector spaces. The equivalence is given by sending $n$ to the vector space $k^n$. A matrix $M \colon n \to m$ is sent to the map $k^n \to k^m, v \mapsto M\cdot v$.
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One of my favorite family of categories are Groups and Groupoids. So for any group G define the category with one object and G the morphisms. Of course there are only finitly many objects, but the category are anything but boring in my opinion, especially if you allow them to be enriched over Top or Mfd. Functor categories out of these categories are a way to categorify representation theory.
But less trivial examples are given by Lie Groupoids and Hopf Algebroids (strictly speaking they are more natural seen as cogroupoids). Lie Groupoids are Groupoids where the Objects and the Union over all Homsets are manifolds and the source and target maps are submersions. They arise naturally in the study of singular foliations. Meanwhile Hopf Algebroids are inner Groupoids in the category of affine schemes. They arise in stable homotopy theory as generalized Steenrodalgebras.
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Since most categories are concrete, the subobjects of an object in a category will usually give rise to a meaningful small category. This, by itself, is not all that interesting, although its failure is: we can construct non-concrete categories by showing that some object in such a category would have too many different subobjects to form a set!
However, a number of related, more interesting instances of essentially small categories arise from topological spaces with various notions of maps. These categories usually fail to be small for trivial reasons (such as "every singleton set is an example, and there are as many singleton sets as there are sets"), but are nonetheless equivalent to some small category. For example, take the category which has compact metric spaces as objects, and isometries as morphisms. This category is essentially small because every compact metric space can be isometrically embedded into a Urysohn universal space.
One needs to take at least some kind of compactness-style condition here, to rule out e.g. discrete spaces of arbitrarily high cardinality. One key result is that any Hausdorff topological space which contains a dense subset of cardinality $\kappa$ has its cardinality bounded above by $2^{2^\kappa}$: this allows us to show that e.g. the category that has separable Hausdorff spaces as objects and continuous maps as morphisms is essentially small.
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I think one should mention complete lattices, posets in which every subset has both a supremum and an infimum. The set of extended real numbers $\mathbb{R} \cup \{+\infty,-\infty\}$ equipped with its usual ordering is probably the most well-known (nontrivial) example.
While degenerate when considered as categories (for any pair of objects $A,B$, we can find at most one morphism from $A$ to $B$, i.e. $|\mathrm{Hom}(A,B)| \leq 1$), they constitute the only examples of complete small categories, i.e. small categories in which all small limits exist.
For if a small category $\mathcal{C}$ has $|\mathrm{Hom}_{\mathcal{C}}(A,B)| \geq 2$ for some objects $A,B \in \mathcal{C}$, the $\#\mathcal{C}$-fold product $B^{\# \mathcal{C}}$, where $\# \mathcal{C}$ denotes the cardinality of the set $\bigcup_{X\in\mathcal{C},Y\in\mathcal{C}} \mathrm{Hom}_{\mathcal{C}}(X,Y)$ of all morphisms of $\mathcal{C}$, necessarily fails to exist: $|\mathrm{Hom}_{\mathcal{C}}(A,B^{\#\mathcal{C}})| \geq 2^{\# \mathcal{C}} > \# \mathcal{C}$ by Cantor's theorem.
NB things are not quite as straightforward in constructive mathematics: see e.g. Hyland's article A small complete category.
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Examples of small categories are abundant.
If a group $G$ acts on a set $X$, then we can form its action groupoid $G//X$. The objects are elements of $X$, and a morphism $x \to y$ is an element $g \in G$ such that $y = gx$. This is a small category. Both orbits and stabilizers are encoded in this category.
For a concrete example, consider the action of $S_3$ on its set of subgroups via conjugation. The action groupoid looks as follows:
Every set can be regarded as a small discrete category (you can also see this as the special case of the action groupoid of the trivial group).
Every preorder can be regarded as a small thin category (meaning there is at most one morphism between any pair of objects). For example, the preorder $(\mathbb{Z},|)$ yields a thin category. This perspective is useful. For example, $\mathrm{gcd}(a,\mathrm{gcd}(b,c)) = \mathrm{gcd}(\mathrm{gcd}(a,b),c)$ (up to a sign) can be derived from the general fact that products in categories are associative (up to isomorphism).
Given any graph, we can consider its path category. The objects are the vertices, the morphisms are the paths, composition of morphisms is concatenation. This is a small category (provided that the graph is small, but usually this is understood).
The fundamental groupoid of a topological space is a small category. Higher categorical generalizations of this construction are also just small categories, since the objects are always just the points of the space.
Groupoids can also be found in puzzle games, namely when the available moves depend on the current position of the game. In Rubik's cube every move can be made in every position, which means that the moves form a group. But for example the moves in the 15 puzzle and the Square-1 from finite groupoids.
Many categories in practice turn out to be essentially small, that is, equivalent to a small category. For example, the category of finitely generated $R$-modules is essentially small, and the category of compact manifolds is essentially small. For many matters of category theory, essentially small categories are just as good as small categories (in some sense even better, since we want to work with properties that are invariant under equivalences of categories).
Another example of this kind is the cobordism category $\mathrm{Cob}(n)$. The objects are (isomorphism classes of) $n$-dimensional closed manifolds, the morphisms are isomorphism classes of cobordisms between these.
Perhaps the most fundamental example of a small category in topology is the simplex category $\Delta$. The objects are non-empty finite linear orders, the morphisms are order-preserving maps. Ok this is only essentially small, which is why one usually works with a small replacement where only linear orders of the form $\{0 < \dotsc < n\}$ are included. This also makes it more feasible to describe the simplex category with generators and relations (the simplicial identities).
There are lots of free small categories (defined by generators and relations) which may be boring at first sight but are very useful nonetheless. For example, there is a category (the walking isomorphism) with two objects and an isomorphism between them, $\{a \leftrightarrow b\}$. It has exactly $4$ morphisms. Given any isomorphism in any category, it comes from this example: there is a functor that maps the universal example to the concrete example. More interestingly, the notion of a reflexive coequalizer can also be described with a small universal example.
Understanding these universal examples is crucial in understanding the generic case, in the same way that polynomials allow us to do ring theory. Moreover, in higher category theory, these small universal examples are not trivial at all, since basically the whole complexity of the subject is already encoded in them, see for example Codescent objects and coherence by Steve Lack.
Any typed programming language yields a small category. The objects are the types, the morphisms are the functions with appropriate argument types and return types. For example, the Python function
def square(x: float) -> float:
return x**2
is an endomorphism of float. Ok usually we want functions to have several arguments, what you get then is called a small multicategory.
Given any theory (in the sense of mathematical logic), we can form its small category of proofs, whose objects are the propositions in that theory and whose morphisms are the proofs showing that one proposition implies the other.
There are small versions of the classical categories you mentioned.
Choose a large set $U$. Usually one assumes $U$ to be closed under taking elements ($M \in U, x \in M \Rightarrow x \in U$) unions, Cartesian products, taking the power set and maybe some more operations. This is not strictly necessary, but ensures that the resulting categories have good properties.
Let $\mathbf{Top}_U$ be the set of all topological spaces $(X, \tau)$, such that $X \in U$. This is really a set since for all $X \in U$, the collection of all topologies on $X$ form a set. Now define Hom exactly as in $\mathbf{Top}$. This makes $\mathbf{Top}_U$ a small category.
You can use the same construction for your the other examples as well.