C. Croke and M. Katz have a number of open problems listed in "Universal volume bounds in Riemannian manifolds" reprint

C. Gordon, P. Perry and D. Schueth have open problems in: "Isospectral and Isoscattering Manifolds: A Survey of Examples" reprint

C. Sormani has a list of problems for graduate students in "Convergence and the Length Spectrum" preprint

Compactness: Is the set of compact Riemannian manifolds isospectral to a given manifold compact in the $C^infty$ sense? Some relevant mathscinet reviews which summerize the status of this question: Brooks-Perry-Petersen, Zhou, Anderson, Chang-Yang, Osgood-Phillips-Sarnak

Gaps in double covers (Sarnak): Let X be a surface with genus g greater than 1 endowed with the hyperbolic metric, and let Y be its universal abelian cover, does the spectrum contain [0, 1/4] or can you create a counter example? This is closely related to the following: Given the same X, does X have a double cover which has the property that all new eigenvalues are greater than or equal to 1/4? This is adapted from the Linial's conjecture that any k regular graph X has a double cover for which all new eigenvalues are in [-2 \sqrt{k-1}, 2\sqrt{k-1}]. This conjecture is based on numerical evidence. Here are transparencies of a talk given by Linial on this issue (see p 45). "A proof that it is false would be interesting, a proof that it is true would be amazing"-Sarnak.