Disclosed are compositions and a method for detecting single nucleic acid molecules using rolling circle amplification (RCA) of single-stranded circular templates, referred to as amplification target circles, primed by immobilized primers. In one form of the method, referred to as a bipartite primer rolling circle amplification, (BP-RCA), RCA of the amplification target circle (ATC) depends on the formation of a primer by target-mediated ligation. In the presence of a nucleic acid molecule having the target sequence, a probe and a combination probe/primer oligonucleotide can hybridize to adjacent sites on the target sequence allowing the probes to be ligated together. By attaching the first probe to a substrate such as a bead or glass slide, unligated probe/primer can be removed after ligation. The only primers remaining will be primers ligated, via the probe portion of the probe/primer, to the first probe. The ligated primer can then be used to prime replication of its cognate ATC. In this way, an ATC will only be replicated if the target sequence (to which its cognate probe/primer is complementary) is present. BP-RCA is useful, for example, for determining which target sequences are present in a nucleic acid sample, or for determining which samples contain a target sequence. In another form of the method, referred to as immobilized primer rolling circle amplification (IP-RCA), RCA of the ATC depends on incorporation of a target sequence in the ATC during its formation. If the target sequence has been incorporated, a primer that can hybridize to the sequence will prime RCA of the ATC. This form of the method is useful for determining which form or forms of a variable sequence are present in a nucleic acid sample.