The existence of Steiner triple systems STS(n) of order n containing no nontrivial subsystem is well known for every admissible n. We generalize this result in two ways. First we define the expander property of 3-uniform hypergraphs and show the existence of Steiner triple systems which are almost perfect expanders. Next we define the strong and weak spreading property of linear hypergraphs, and determine the minimum size of a linear triple system with these properties, up to a small constant factor. This property is strongly connected to the connectivity of the structure and of the so-called influence maximization. We also discuss how the results are related to Erdős' conjecture on locally sparse STSs, influence maximization, subsquare-free Latin squares and possible applications in finite geometry.