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I believe that the problem of characterizing the sets of divergence for classical Fourier series is more or less open for all interesting classes ($C$, $L^\infty$, $L^p$ with $p>1$).

The strongest result that I'm aware of is due to Buzdalin who showed that any null-set $E\in F_\sigma\cap G_\delta$ is a set of divergence for the Fourier series of some continuous complex-valued function ("Trigonometric Fourier series of continuous functions diverging on a given set", Math. USSR Sbornik, 24 (1974)).

The characterization problem is mostly solved however for several other orthogonal systems, including the Haar and Franklin systems. There is also a very recent paper by Karagulyan where it is proved, in particular, that

A necessary and suﬃcient condition for a set $E \subset [0, 1]$ to be a set of divergence for the sequence of $(C, \alpha)$-means ($\alpha>0$) of the Fourier series for of some function $f \in L^\infty[0, 1]$ is that $E$ is a $G_{\delta\sigma}$-set of measure $0$.

(See G.A. Karagulyan, "Characterization of the sets of divergence for sequences of operators with the localization property", Sbornik: Mathematics, 202 (2011), pp. 9–33.)

To complicate things further, people tend to distinguish between the sets of divergence and unbounded divergence. A set $E \subset [0, 1]$ is said to be a set of divergence (resp. unbounded divergence) for a series of functions $$\sum_{n=1}^{\infty}f_n(x),\qquad x\in[0,1],$$ if the series diverges for $x E$ and converges for $x \in [0, 1] \backslash E$ (resp. diverges unboundedly for $x E$).

One may think of the two optimistic working conjectures.

1. Every $G_{\delta\sigma}$-set $E$ of measure $0$ is a set of divergence for the Fourier series of some function $f \in C[0, 1]$.

2. Every $G_{\delta}$-set $E$ of measure $0$ is a set of unbounded divergence for the Fourier series of some function $f \in C[0, 1]$.

Conjecture 2 was explicitly formulated by P.L. Ul'yanov in the late 1960s. Both conjectures seem to be open.

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I believe that the problem of characterizing the sets of divergence for classical Fourier series is more or less open for all interesting classes ($C$, $L^\infty$, $L^p$ with $p>1$).

The strongest result that I'm aware of is due to Buzdalin who showed that any null-set $E\in F_\sigma\cap G_\delta$ is a set of divergence for the Fourier series of some continuous complex-valued function ("Trigonometric Fourier series of continuous functions diverging on a given set", Math. USSR Sbornik, 24 (1974)).

The characterization problem is mostly solved however for several other orthogonal systems, including the Haar and Franklin systems. There is also a very recent paper by Karagulyan where it is proved, in particular, that

A necessary and suﬃcient condition for a set $E \subset [0, 1]$ to be a set of divergence for the sequence of $(C, \alpha)$-means ($\alpha>0$) of the Fourier series for some function $f \in L^\infty[0, 1]$ is that $E$ is a $G_{\delta\sigma}$-set of measure $0$.

(See G.A. Karagulyan, "Characterization of the sets of divergence for sequences of operators with the localization property", Sbornik: Mathematics, 202 (2011), pp. 9–33.)