The recent development and application of methods based on the general principle of "crosslinking and proximity ligation" (crosslink-ligation) are revolutionizing RNA structure studies in living cells. However, extracting structure information from such data presents unique challenges. Here, we introduce a set of computational tools for the systematic analysis of data from a wide variety of crosslink-ligation methods, specifically focusing on read mapping, alignment classification, and clustering. We design a new strategy to map short reads with irregular gaps at high sensitivity and specificity. Analysis of previously published data reveals distinct properties and bias caused by the crosslinking reactions. We perform rigorous and exhaustive classification of alignments and discover eight types of arrangements that provide distinct information on RNA structures and interactions. To deconvolve the dense and intertwined gapped alignments, we develop a network/graph-based tool Crosslinked RNA Secondary Structure Analysis using Network Techniques (CRSSANT), which enables clustering of gapped alignments and discovery of new alternative and dynamic conformations. We discover that multiple crosslinking and ligation events can occur on the same RNA, generating multisegment alignments to report complex high-level RNA structures and multi-RNA interactions. We find that alignments with overlapped segments are produced from potential homodimers and develop a new method for their de novo identification. Analysis of overlapping alignments revealed potential new homodimers in cellular noncoding RNAs and RNA virus genomes in the Picornaviridae family. Together, this suite of computational tools enables rapid and efficient analysis of RNA structure and interaction data in living cells.