This thesis presents elastic moduli measurements in two materials, the heavy-fermion material URu$_2$Si$_2$ and the unconventional superconductor Sr$_2$RuO$_4$. Both of these materials have phase transitions where, in spite of extensive experimental efforts, the exact broken symmetries have remained unknown. Our measurements place strong thermodynamic constraints on the order parameter (OP) symmetry at these transitions. I start by explaining how measurements of elastic moduli discontinuities at a second order transition can help constrain or identify the symmetries of the OP that condenses. Then I describe in details the experimental technique used to measure these moduli discontinuities, resonant ultrasound spectroscopy (RUS). The first material I studied, URu$_2$Si$2$, has a "hidden order" transition at $T\mathrm{HO}=17.5$ K, with multiple proposals for the OP. The results, published in \textit{Science Advances}, 6(10), 2020 [1], rules out theories of hidden order that are based on two-component OPs. The second material, Sr$_2$RuO$4$, becomes a superconductor at $T\mathrm{c}=1.45$ K. The measurements in Sr$_2$RuO$_4$ provide evidence for a two-component OP, and narrows down the possibilities to two specific order parameter candidates. These were published in \textit{Nature Physics}, 17(2):199–204, Feb 2021 [2]. Further, ultrasound attenuation in Sr$_2$RuO$4$ is seen to immensely increase below $T\mathrm{c}$, thereby signifying highly anomalous dynamics. Formation of superconducting domains is found to best explain this striking observation, as detailed in this arXiv preprint [3].