The scattering of waves by a finite thin plate in a two-dimensional wave guide and an array of finite thin plates, in the presence of subsonic mean flow, are formulated using a mode matching technique. The influence of mean flow on trapped modes in the vicinity of a finite thin plate in a two-dimensional wave guide is then investigated by putting the amplitude of the forcing term to zero in the scattering problem. The conditions for complex resonances are found, and numerical results are computed. The influence of mean flow on Rayleigh-Bloch modes is investigated by using a similar methodology. The condition for embedded trapped modes to exist is introduced next, and then numerical results for embedded trapped modes without mean flow are presented. Complex resonances without mean flow are then found by fixing the geometry of the waveguide. The influence of mean flow on complex resonances and embedded trapped modes is investigated subsequently. In addition, the investigation of scattering coefficients is discussed when the frequency of an incident wave is near the real part of the frequency of complex resonances or embedded trapped modes. Embedded trapped modes near an indentation in a strip wave guide, which may correspond to a two-dimensional acoustic wave guide or a channel of uniform water depth in water waves, are also found. Modes are sought which are either symmetric or anti-symmetric about the centreline of the guide and the centre of the indentation. In each case, a simple approximate solution is found numerically. Full solutions are then found by using a Galerkin approach in which the singularity near the indentation edge is modelled by choosing proper special functions. The final part of the thesis is devoted to spinning modes (Rayleigh-Bloch modes) in a cylindrical waveguide in the presence of radial fins. A mode matching technique is used to obtain the potential, and the coefficients in the expansion are found numerically by using an efficient Galerkin procedure. In addition, an existence proof for modes symmetric about the centre of the guide and the centre of the section with radial fins is given by applying a variational approach. The connection between Rayleigh-Bloch modes and trapped modes is discussed thereafter, and numerical results for a number of geometric configurations are presented.