Flows around immersed boundaries exhibit many complex, well defined and active dynamical structures. In fact, features such as shock waves, strong vorticity concentrations in shear layers, wakes, or boundary layer regions are critical elements in representing the dynamics of a flow field. In order to capture the correct kinematic and dynamic quantities associated with the fluid flows, one must be able to efficiently refine the computational mesh around areas containing high gradients of pressure, density, velocity, or other suitable flowfield variables that characterize distinct structures. Although there are techniques which utilize simple gradient-based Local Mesh Refinement (LMR) to adapt resolution selectively to capture structures in the flow, such methods lack the ability to refine structures based on the relative strengths and scales of structures that are presented in the flow. The inability to adequately define the strength and scale of structures typically results in the mesh being over-refined in regions of little consequence to the physical definition of the problem, under-refined in certain regions resulting in the loss of important features, or even the emergence of false features due to perturbations in the flowfield caused by unnecessary mesh refinement. On the other hand, significant user judgment is required to develop a "good enough" mesh for a given flow problem, so that important structures in the flowfield can be resolved. In order to overcome this problem, multiresolution techniques based on the wavelet transform are explored for feature identification and refinement. Properties and current uses of these functional transforms in fluid flow computations will be briefly discussed. A Multiresolution Transform (MRT) scheme is chosen for identifying coherent structures because of its ability to capture the scale and relative intensity of a structure, and its easy application on non-uniform meshes. The procedure used for implementation of the MRT on an octree/quadtree LMR mesh is discussed in detail, and techniques used for the identification and capture of jump discontinuities and scale information are also presented. High speed compressible flow simulations are presented for a number of cases using the described MRT LMR scheme. MRT based mesh refinement performance is analyzed and further suggestions are made for refinement parameters based on resulting refinement. The key contribution of this thesis is the identification of methods that lead to a robust, general (i.e. not requiring user-defined parameters) methodology to identify structures in compressible flows (shocks, slip lines, vertical patterns) and to direct refinement to adequately refine these structures. The ENO-MRT LMR scheme is shown to be a robust, automatic, and relatively inexpensive way of gaining accurate refinement of the major features contained in the flowfield.