The present invention provides methods of making and using self-assembled arrays of single polynucleotide molecules for carrying out a variety of large-scale genetic measurements, such as gene expression analysis, gene copy number assessment, and the like. Random arrays used in the invention are “self-assembled” in the sense that they are formed by deposition of polynucleotide molecules onto a surface where they become fixed at random locations. The polynucleotide molecules fixed on the surface are then identified by direct sequence determination of component nucleic acids, such as incorporated probe sequences, or by other decoding schemes. Such identification converts a random array of determinable polynucleotides, and their respective probes into an addressable array of probe sequences.