<p>A staggered linear basis (SLB) provides a particular linear basis for the ideal it generates and contains a Gröbner basis for this ideal. These properties are enhanced through the additional structure of sets of allowed and forbidden terms assigned to each polynomial within the SLB. In the first part of this paper, we develop the theory of SLBs by introducing and studying <i>minimal</i> staggered linear bases, and by presenting an algorithm for their computation. The second part of the paper explores several applications of SLBs, including the computation of Hilbert functions, Hilbert polynomials, and Hilbert series for polynomial ideals. Furthermore, by leveraging the combinatorial structure of SLBs, we introduce an algorithm for constructing irreducible complementary decompositions for a given monomial ideal. Finally, we present several algorithms that address various aspects of complementary decompositions of ideals. We conclude the paper by reporting the first implementation of SLBs in the <Emphasis FontCategory="NonProportional">CoCoALib</Emphasis>. We compare its efficiency with that of the built-in function for Gröbner basis computations, showing that for some classes of examples and coefficient fields, our algorithm outperforms the built-in one.</p>