I will explain how the Selberg trace formula can be used to prove upper bounds on the eigenvalues of the Laplacian and their multiplicities or on the lengths of closed geodesics and their multiplicities, for arbitrary closed hyperbolic surfaces. For surfaces with a large automorphism group, we can use a version of the trace formula adapted to its irreducible representations to prove rigorous lower and upper bounds on the Laplace spectrum. This method yields counterexamples to an upper bound on the multiplicity of the first nonzero Laplacian eigenvalue conjectured by Colin de Verdière in 1986. This is joint work with Émile Gruda-Mediavilla, Bram Petri, and Mathieu Pineault.