Significant advances in compressive sensing and sparse signal encoding have provided a rich set of mathematical tools for signal analysis and representation. In addition to novel formulations for enabling sparse solutions to underdetermined systems, exciting progress has taken place in efficiently solving these problems from an optimization theoretic viewpoint. The focus of the wide body of literature in compressive sensing/sparse signal representations has however been on the problem of signal recovery from a small number of measurements (equivalently a sparse coefficient vector). This tutorial will discuss the design of sparse signal representations explicitly for the purposes of signal classification. The tutorial will focus on and build upon two significant recent advances. First, the work by Wright et al. which advocates the use of a dictionary (or basis) matrix comprising of class-specific training sub-dictionaries. In this framework, a test signal is modeled as a sparse linear combination of training vectors in the dictionary, sparsity being enforced by the assertion that only coefficients corresponding to one class (from which the test signal is drawn) ought to be active. The second set of ideas we leverage are recent key contributions in model-based compressive sensing where prior information or constraints on sparse coefficients are used to enhance signal recovery. These ideas will be combined towards the exposition of current trends: namely the development of class-specific priors or constraints to capture structure on sparse coefficients that helps explicitly distinguish between signal classes. In the second part of the tutorial, applications will be discussed including: 1.) structured sparsity for classification of medical imagery for diagnostics, 2.) low-rank approximation and sparse recovery for visual data reconstruction, and 3.) sparse representations for target detection and classification in hyperspectral imagery (guest speaker from the US Army Research Lab).