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We investigate the optical properties of microcavities with suspended subwavelength structured mirrors, such as high-contrast gratings or two-dimensional photonic crystals slabs, and focus in particular on the regime in which the microcavity free-spectral range is larger than the width of a Fano resonance of the highly reflecting structured mirror. In this unusual regime, the transmission spectrum of the microcavity essentially consists in a single mode, whose linewidth can be significantly narrower than both the Fano resonance linewidth and the linewidth of an equally short cavity without structured mirror. This generic interference effect---occuring in any Fabry-Perot resonator with a strongly wavelength-dependent mirror---can be exploited for realizing small modevolume and high quality factor microcavities and, if high mechanical quality suspended structured thin films are used, for optomechanics and optical sensing applications., 4 pages, 5 figures