A famous question of Mark Kac asks whether one can "hear the shape of a drum," that is, whether a domain can be recovered from the spectrum of its Laplacian. Its dynamical counterpart asks whether a Birkhoff billiard table can be determined by the collection of lengths of its periodic orbits. The answer to this question is closely related to the complexity of the length spectrum. In particular, one may ask whether "pathological" length spectra occur. We show that this is a dense phenomenon: the set of Birkhoff billiard tables with uncountable length spectrum is dense in the space of billiard tables. To prove this result, we develop a local perturbation theory for Birkhoff billiards, with broad potential applications to billiards in general. Based on joint work with J. De Simoi and Ke Zhang