And are the eigenfunctions countable?
This question was motivated thinking about operators in quantum mechanics, which are Hermitian, however I would be interested in knowing about operators in general.
Some of the operators I have come across in Quantum mechanics which have infinitely many eigenvalues are the position and momentum operators (generally uncountably infintely many eigenvalues and eigenstates), the Hamiltonian which can have uncountably infinite energy spectrum for an unbound particle, or countably infinite for a bound particle. It seems that the operators which have discrete eigenvalues necessarily have countable ones. I was wondering if this is a general result.
A discrete subset of $\mathbb{C}$ is countable (either finite or infinite), yes. As for the eigenfunctions, technically you could have discrete spectrum and still have uncountably many linearly independent eigenfunctions, just consider the zero operator on a space of large enough dimension. But this is a rather pathological case.