While thinking about whether the fock space of a free scalar field is separable, I distilled the initial question into the following problem.
Consider the one-particle hilbert space for a scalar field. If we go through the motions of solving the free scalar-field equation, we will find that the creation and annihilation operators are indexed by $k$, let us say for simplicity it is a real number describing the momentum of the particle being created. Knowing that, then the one-particle Hilbert space should be the completion of the vector space spanned by the vectors : $$ \left|k\right> = a_k^\dagger\left|0\right> $$ Where $a_k^\dagger$ is the creation operator and $\left|0\right>$ is the vacuum, defined as the state or vector being annihilated by all $a_k$'s. I already have a problem with what should be the associated scalar product. I guess it should be inherited from the relations $\left<k'\right|\left|k\right>\propto\delta(k-k')$, but I don't really know what to do with the delta function.
Now, I would like to find a countable Hilbert basis for this Hilbert space (the $\left|k\right>$ are clearly a Hilbert basis, but uncountable). Namely I would like to find a countable set of vectors $\left|i\right>$ such that their span is dense in the Hilbert space, w.r.t. to the above norm (which admittedly I am not sure how to define).
So to summarize : is the Hilbert space for a single particle free scalar field even well-defined ? If yes, is it separable (which implies iff that there is a countable Hilbert basis)?