This work explores sign regular (SR) matrices, a key class in matrix theory with applications in mathematics, statistics, engineering, economics, and computer design. A SR matrix is defined as a matrix whose all minors of the same order with identical signs, with specific subclasses determined by the matrix signature. Among these, totally positive (TP) and totally negative (TN) matrices are particularly important. We investigate the properties and subclasses of SR matrices, focusing on their connection to combined matrices, checkerboard matrices, and doubly quasi-stochastic matrices. A central objective is to determine if there exists a nonsingular matrix in specific SR subclasses whose combined matrix is both doubly quasi-stochastic and strictly checkerboard or strictly inverse checkerboard. The chapter introduces key concepts and presents new results, including a characterization of these matrices based on their signatures. Theoretical results are grounded in existing literature and extended through novel constructions. Further chapters provide detailed exploration of specific SR subclasses and their properties, ensuring a comprehensive analysis of these structured matrices.