Basis in quantum mechanics

First, the term "basis spaces" isn't standard in quantum physics but let us assume that we understand what the sentences approximately mean.

Second, the momentum is continuous (not quantized) if the position space is noncompact (infinite). The momentum only becomes quantized if the position space is compact (or periodic), and indeed, it's been experimentally verified that the momentum is quantized e.g. in the potential well.

Third, the same Hilbert space (more precisely, rigged Hilbert space etc.) may have both countable and uncountable (labeled by continuous numbers) bases. There is no contradiction because at this level of accuracy, the cardinality of the basis doesn't have an impact on the "size" of the Hilbert space as long as it is infinite-dimensional. See e.g.

http://motls.blogspot.com/2014/02/cardinality-of-bases-doesnt-matter-for.html?m=1

So for example, a particle on the line is described by the Hilbert space of complex-valued functions $\psi(x)$ which may be constructed out of the continuous "bases" of position eigenstates; or as superpositions of the harmonic oscillator energy eigenstates (this basis is countable). From the viewpoint of a mathematician who likes to think about cardinality of sets, the bases may be "differently large". But from the viewpoint of physics, they are equally large.

It is completely common and normal in quantum mechanics that some operators have continuous spectra, other operators have discrete spectra, and other operators have mixed (discrete plus continuous) spectra. They may still have bases of eigenstates – which are exactly sufficient to write every vector as a linear superposition. For the discrete bases, the linear superposition is written as a sum; for the continuous bases, the linear combination is written as an integral (and has to be supported by some extended axiomatic system of "rigged Hilbert spaces" etc. to remain rigorous); for mixed bases, the superposition is the sum of the sum and an integral.

Operators with continuous, discrete, and mixed spectra must be considered "equally good operators" from the physics viewpoint. Indeed, the Hamiltonian – the operator of energy that also governs the time evolution – may have discrete, continuous, or mixed spectra and the answer often requires complicated dynamical calculations: it is in no way determined "a priori" whether the Hamiltonian should have a discrete part of the spectrum. Its spectrum has a discrete part (the spectrum is discrete or mixed) if the Hamiltonian admits "bound states" and it usually can't be guessed "immediately" whether such bound states exist.