This is a very interesting question. First, it is not at all obvious how (or whether it is possible) to extend the relatively elementary Fourier-Dirichlet result on pointwise convergence to general orthonormal systems (let's say even for (the simplest, "non-singular") Sturm-Liouville problems with smooth coefficients). Yes, of course, the $L^2$ convergence is the main point, and for utilitarian purposes not so many people ask further. And, yes, everyone imagines that for very smooth functions, the spectral expansion in terms of the eigenfunctions should converge pointwise, etc.
It cannot be quite this simple, unfortunately/interestingly enough. For example, the eigenfunctions for the Dirichlet problem $u''=\lambda u$ and $u(0)=0=u(2\pi)$ are $\sin(nx/2)$. So of course things like the constant function $1$ are in their $L^2$ span. But that $L^2$ expansion cannot possibly converge pointwise to $1$, because $1$ is just $1$, at the endpoints, while all the eigenfunctions are $0$ there.
Yes, this failure can be viewed as essentially irrelevant. And, indeed, for functions in the corresponding $H^1$ Sobolev space (attached to the operator + boundary conditions), since we have Sobolev imbedding $H^1\subset C^o$, convergence is in $C^o$.
But, still, the eigenfunctions themselves are only in $H^1$, not $H^\infty$. So there will inevitably be troubles... understandably, at the endpoints, but such troubles propagate in spectral expansions (for the same reasons that Fourier transforms interchange local smoothness and global decay).