Molecular recognition elements (MREs) can be short sequences of single-stranded DNA, RNA, small peptides, or antibody fragments. They can bind to user-defined targets with high affinity and specificity. There has been an increasing interest in the identification and application of nucleic acid molecular recognition elements, commonly known as aptamers, since they were first described in 1990 by the Gold and Szostak laboratories. A large number of target specific nucleic acids MREs and their applications are currently in the literature. This review first describes the general methodologies used in identifying single-stranded DNA (ssDNA) aptamers. It then summarizes advancements in the identification and biosensing application of ssDNA aptamers specific for bacteria, viruses, their associated molecules, and selected chemical toxins. Lastly, an overview of the basic principles of ssDNA aptamer-based biosensors is discussed.