We focus on $2\pi$-periodic functions.
1- Does there exist any differentiable function whose Fourier series is not uniformly convergent?
Even more,
2- Does there exist any differentiable function whose Fourier series con not be uniformly convergent on every set of positive measure in [$-2\pi,2\pi$]?
Perhaps you already know the two things I have to say about this: (i) Dini's test shows that if $f$ is differentiable then the Fourier series fo $f$ converges pointwise to $f$ (ii) it follows, and it's also easy to prove directly, that if $f$ is continuously differentiable then the Fourier series converges to $f$ uniformly. (Because $f'$ continuous implies $\sum n^2|c_n|^2<\infty$, hence $\sum|c_n|<\infty$.)
But if $f$ is just differentiable does that imply the Fourier series for $f$ converges uniformly? I haven't concocted an example, but surely not; if $f'$ exists but is not continuous then the differentiation is not uniform.