One thing that's always bothered me in Dirac's notation is that it assumes that the Hilbert space contains a "continuum basis" of vectors $|x\rangle$, which happen to be eigenvectors of an operator $X$ (which has no eigenvalues, only a continuous spectrum that spans the whole space). Their inner product is distribution-valued, with $\langle x'|x\rangle = \delta(x'-x)$. There's also the cryptic normalization property: $\int |x\rangle\langle x| dx = Id$. According to this question, some of these can't be made rigorous even with something like the concept of "Rigged Hilbert Spaces".
So, is there another approach to quantum mechanics in general that sidesteps this issue entirely, without loss of descriptive power?