In this thesis we use various techniques to investigate the occurrence of trapped modes in the presence of thin obstacles. Physically trapped modes are oscillations of finite energy in a fluid which is unbounded in at least one direction. These oscillations mainly occur locally to some structure and decay to zero at large distances away from it. Trapped modes are important as they have been found to exist in a wide range of physical situations. We consider a number of problems in two and three dimensions including waveguides containing bodies and arrays of identical structures. A modified residue calculus technique, a variational technique and a method based on the truncation of matched eigenfunction expansions are used to solve the problems, with numerous results presented.