The sensitivity of the resonant optical sensors, which are based on measurement of the transmission and reflection spectra of optical resonators, is investigated. The following problem is addressed: When the losses of the resonator are known, what is the sharpest possible and the steepest possible shape of the resonant peaks that can be achieved experimentally? This optimization problem is solved for the case of a separated peak, which corresponds to a nondegenerated eigenvalue of the resonator. It is shown that the reflection spectrum possesses better sensitivity than the transmission spectrum. The model of the resonant sensor consisting of two coupled resonators is also considered. This model demonstrates that the sensitivity of transmission spectrum can be significantly increased by modification of the resonator structure. However, for the reflection spectrum, the best sensitivity is still given by a separated resonant peak.