Techniques in frequency stabilization of lasers to fixed-spacer optical cavities have advanced to the point where the ultimate frequency stabilities are limited by thermal noise in the cavity materials for standard cavity configurations at room temperature. The use of spectral-hole burning (SHB) in laser stabilization has produced promising results in early experiments. In this letter we explore the thermal-noise limits to frequency stability in burned spectral holes. We compile known material parameters for a typical system used in SHB experiments (Eu 3+ doped Y 2 SiO 5 ) to make numerical estimates for the fundamental thermal-noise induced frequency instability in spectral-holes for the liquid-helium temperature and dilution temperature cases. These efforts can guide the design of future SHB experiments and clarify which important material parameters remain to be measured.