Non-sequential spaces in the wild TLDR: What are examples of (function-)spaces that are not sequential? When does this matter?
As a simple analyst, I am most happy if I can just work with sequences all the time. In most situations this is totally fine, as many many spaces one encounters in one's daily life are actually sequential  (or even 1st countable, or even better metrisable). Now recently I was a bit shocked to find out that the seemingly familiar space of test-functions $\mathscr{D}(\mathbb{R}^d)$ (with its usual LF-topology) actually fails to be sequential. But hadn't I learned that one can verify whether a linear functional on $\mathscr{D}(\mathbb{R}^d)$ is a distribution by checking continuity with sequences? Well in that case it is true (Proposition 21.1 in Trèves TVS book), but only because we looked at linear functionals.
This got me thinking that there might actually be a bunch of spaces around, not pathological counterexamples, but real spaces one encounters in the wild, that fail to be sequential. In some cases, like above, this might not be problematic, but potentially for non-trivial reasons. In order to become more aware of these subtleties I would like to collect some more examples of this.
An answer should ideally contain the following:

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*A concrete example or a class of examples of non-sequential spaces, which are widely used or naturally show up in analysis. My main interested lies in topological vector spaces that appear as function spaces in some context. The example should not be some 'patholocial counterexample' (this is of course a bit vague).

*An instance of where it matters that the space is non-sequential. Or a warning, of when one needs to be more careful with it and use filters or nets.

*Loop holes or special situations where it suffices to focus on sequences nevertheless.

I'll make a start:

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*Test-functions: $\mathscr{D}(\mathbb{R}^d)$ (with standard LF-topology)  is not sequential. In particular a function $f:\mathscr{D}(\mathbb{R}^d)\rightarrow \mathbb{C}$ might be sequentually continuous, but not continuous (example by PhoemueX). However, if $f$ is linear then sequential continuity implies continuity (Corollary after Proposition 13.1 in Trèves' TVS book). The same is true for other LF-spaces.

*Distributions: $\mathscr{D}'(\mathbb{R}^d)$ (with the strong topology) is not sequential. A sequence of distributions converges strongly if and only if it converges weakly, but this is not true when sequences are replaced by nets/filters. The same result (for sequences) holds in strong duals of Montel spaces (Corollary 1 to Proposition 34.6 in Trèves)

*An infinite dimensional Banach space, equipped with the weak topology is not sequential. However, despite this we have that compactness = sequential compactness (Eberlein-Smulian theorem).

Finally, here are some spaces that are sequential: $\mathscr{S}(\mathbb{R}^d)$ (Schwartz-space), $\mathscr{D}(M)$ (distributions on compact manifold $M$), the dual of a separable locally convex space with the weak$^*$-topology, ...
 A: The unit ball of the dual of a separable normed space is weak$^*$ sequentially compact, but this fail dramatically for non-separable spaces: The sequence of evaluations $\delta_n: \ell_\infty\to\mathbb R$ is probably the first sequence in $\ell^*_\infty$ that comes to mind and it has no weak$^*$-convergent subsequence: This would be a sequence of integers $n_1<n_2<\cdots$ such that for every bounded sequence $(x_n)_{n\in\mathbb N}$ of scalars one had a limit $\lim\limits_{k\to\infty} x_{n_k}$.
A: The sequentiality does not match well with an algebraic structure. For example, the following result of Banakh and Zdomskyy characterizes sequential topological groups with countable $cs^*$-character:

Theorem. A topological group $G$ with countable $cs^*$-character is sequential if and only if $G$ is either metrizable or contains an open $\mathcal M\mathcal K_\omega$-subgroup.

Let us recall that a topological space $X$ has countable $cs^*$-character if for every point $x\in X$ there exists a countable family $\mathcal F_x$ of subsets of $X$ such that for every neighborhood $O_x\subseteq X$ of $x$ and every sequence $\{x_n\}_{n\in\omega}\subseteq X$ that converges to $x$, there exist a set $F\in\mathcal F_x$ such that $F\subseteq O_x$ and $F$ contains infinitely many points of the sequence $(x_n)$.
A topological space $X$ is $\mathcal{MK}_\omega$ if there exists a countable cover $\mathcal C$ of $X$ by compact metrizable subspaces such that a subset $F\subseteq X$ is closed if and only if for every compact set $C\in\mathcal C$ the intersection $C\cap F$ is closed in $C$.
In this paper of Banakh and Repovs the above result of Banakh--Zdomskyy was extended to rectifiable spaces and topological left-loops.
In fact, the above theorem, is a corollary of the following result of Banakh:

Theorem. If a perfectly normal topological group $G$ contains a topological copy of the Frechet-Urysohn fan $S_\omega$ and a closed topological copy of the metric fan $M$, then $G$ is not sequential.

The metric fan is the subspace $$M=\{0\}\cup\{\tfrac1{n}+\tfrac{i}{nm}:n,m\in\mathbb N\}$$ of the complex plane. The Fr'echet-Urysohn fan is the set $M$ endowed with the strongest topology that coincides with the Euclidean topology on each subspace $K_m=\{0\}\cup\{\frac1{n}+\tfrac{i}{nm}:n\in\mathbb N\}$, $m\in\mathbb N$. It is easy to see that the Fr'echet-Urysohn fan is an $\mathcal{MK}_\omega$-space.
