We present a method for finding quasi-resonances for a linear acoustic transmission problem in the frequency domain. Starting from an equivalent boundary-integral equation we perform Galerkin boundary element discretization and look for the minima of the smallest singular value of the resulting matrix as a function of the wave number $k$. From the perspective of the numerical methods for spectral problems, we present error estimates and an adaptive algorithm for approximating the solution of a particular parametric eigenvalue problem. We study the parameter dependent family of singular value problems. To justify the proposed algorithm, we present new error estimates for the accuracy of the Galerkin discretization of singular values of Fredholm operators. We also present heuristic adaptive algorithm for finding the minima in prescribed $k$-intervals.