Any function that is in $L^1$ but not in $L^\infty$ will have some point in whose neighbourhood it is unbounded, and the Fourier series is likely to diverge at such a point. A simple example is $$ f(x) = \ln(1-\cos(x)) = -\ln(2) + \sum_{n=1}^\infty \dfrac{2 \cos(nx)}{n} $$ where the series diverges at multiples of $2\pi$.

Any function that is in $L^1$ but not in $L^\infty$ will have some point in whose neighbourhood it is unbounded, and the Fourier series is likely to diverge at such a point. A simple example is $$ f(x) = \ln(1-\cos(x)) = -\ln(2) + \sum_{n=1}^\infty \dfrac{2 \cos(nx)}{n} $$ where the series diverges at multiples of $2\pi$.

EDIT: For Fejér's example of a continuous function whose series diverges at a point, see e.g. Edwards, "Fourier series", sec. 10.3.1.