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Spontaneous symmetry breaking is ubiquitous in physics. Its spectroscopic signature consists in the softening of a specific mode upon approaching the transition from the high symmetry side and its subsequent splitting into a zero-frequency \"Goldstone\" mode and a non-zero-frequency \"Higgs\" mode. Although they determine the whole system dynamics, these features are difficult to address in practice because of their vanishing coupling to most experimental probes and/or their strong interaction with other fluctuations. In this work, we consider a periodic magnetic modulation occurring in a ferromagnetic film with perpendicular-to-plane magnetic anisotropy and directly observe its Goldstone and Higgs spin-wave modes at room temperature using microwave and optical techniques. This simple system constitutes a particularly convenient platform for further exploring the dynamics of symmetry breaking.