Thermal noise in optical cavities imposes a severe limitation in the stability of the most advanced frequency standards at a level of a few 10^-16 √s/τ for long averaging times τ. In this paper, we describe two schemes for reducing the effect of thermal noise in a reference cavity. In the first approach, we investigate the potential and limitations of operating the cavity close to instability, where the beam diameter on the mirrors becomes large. Our analysis shows that even a 10-cm short cavity can achieve a thermal-noise-limited fractional frequency instability in the low 10⁻¹⁶ regime. In the second approach, we increase the length of the optical cavity. We show that a 39.5-cm long cavity has the potential for a fractional frequency instability even below 10⁻¹⁶, while it seems feasible to achieve a reduced sensitivity of <10⁻¹⁰/g for vibration-induced fractional length changes in all three directions.