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][1]", sec. 10.3.1. [1]: https://books.google.ca/books?id=LsDkBwAAQBAJ