Every observable in the technical or mathematical sense (linear Hermitian operator on the Hilbert space) is, in principle, observable in the physical operational sense, too. That's why it's called this way.
Magnetic fields may be measured, for example, by compasses. Analogous methods exist for electric fields, scalar fields, or any other fields. For example, if you want to measure the Higgs field, you may, in principle, place a top quark (or an even heavier particle if there is one) at that point and measure its induced inertial mass.
Let me mention that a true observable must be gauge-invariant. So if a complex field carries a charge $Q$, it is not gauge-invariant. One has to combine it to expressions such as $\phi^\dagger \phi$ to get gauge-invariant objects. These are true observables. This extra requirement doesn't contradict the original definition because gauge-non-invariant operators are not well-defined linear operators acting on the physical Hilbert space (because physical states are equivalence classes and the action of a gauge-non-invariant operator would depend on the representative of the class). Yes, by the Hilbert space, I always meant the physical ones, after all identifications that should be made are made and unphysical states such as longitudinal photons are removed.
Also, fermionic fields may be called observables but they can't have nonzero eigenvalues. Only products that are Grassmann-even – contain an even number of fermionic factors – are measurable due to the existence of superselection sectors that divide bosonic and fermionic states according to the eigenvalue of $(-1)^F$. But formally speaking, we could imagine states in the Hilbert space with Grassmann-odd coefficients and the "fermionic coherent states" would be eigenvalues of fermionic operators. However, Grassmann-odd probability amplitudes aren't physical so such a construction is purely formal.