Certain scalar-tensor theories have the property of endowing stars with scalar hair, sourced either by the star's own compactness (spontaneous scalarization) or, for binary systems, by the companion's scalar hair (induced scalarization) or by the orbital binding energy (dynamical scalarization). Scalarized stars in binaries present different conservative dynamics than in general relativity, and can also excite a scalar mode in the metric perturbation that carries away dipolar radiation. As a result, the binary orbit shrinks faster than predicted in general relativity, modifying the rate of decay of the orbital period. In spite of this, scalar-tensor theories can pass existing binary pulsar tests, because observed pulsars may not be compact enough or sufficiently orbitally bound to activate scalarization. Gravitational waves emitted during the last stages of compact binary inspirals are thus ideal probes of scalarization effects. For the standard projected sensitivity of advanced LIGO, we here show that, if the neutron star equation of state is such that the stars can be sufficiently compact that they enter the detector's sensitivity band already scalarized, then gravitational waves could place constraints at least comparable to binary pulsars. If the stars dynamically scalarize while inspiraling in band, then constraints are still possible provided the equation of state leads to scalarization that occurs sufficiently early in the inspiral, roughly below an orbital frequency of 50 Hz. In performing these studies, we rederive an easy-to-calculate data analysis measure, an integrated phase difference between a general-relativistic and a modified signal, and connect it directly to the Bayes factor, showing that it can be used to determine whether a modified gravity effect is detectable. Finally, we find that custom-made templates are equally effective as model-independent, parametrized post-Einsteinian waveforms at detecting such modified gravity effects at realistic signal-to-noise ratios.