A linear graph is a graph whose vertices are linearly ordered. This linear ordering allows pairs of disjoint edges to be either preceding (<), nesting (⊏) or crossing (≬). Given a family of linear graphs, and a non-empty subset R⊆{<,⊏,≬}, we are interested in the Maximum Common Structured Pattern (MCSP) problem: find a maximum size edge-disjoint graph, with edge pairs all comparable by one of the relations in R, that occurs as a subgraph in each of the linear graphs of the family. The MCSP problem generalizes many structure-comparison and structure-prediction problems that arise in computational molecular biology.We give tight hardness results for the MCSP problem for {<,≬}-structured patterns and {⊏,≬}-structured patterns. Furthermore, we prove that the problem is approximable within ratios: (i) 2ℋ(k) for {<,≬}-structured patterns, (ii) k1/2 for {⊏,≬}-structured patterns, and (iii) O(klogk) for {<,⊏,≬}-structured patterns, where k is the size of the optimal solution and ℋ(k)=∑i=1k1/i is the kth harmonic number. Also, we provide combinatorial results concerning different types of structured patterns that are of independent interest in their own right.