The use of CRISPR/Cas nucleases for the development of DNA diagnostic systems in out-of-laboratory conditions (point-of-need testing, PONT) has demonstrated rapid growth in the last few years, starting with the appearance in 2017–2018 of the first diagnostic platforms known as DETECTR and SHERLOCK. The platforms are based on a combination of methods of nucleic acid isothermal amplification with selective CRISPR/Cas detection of target amplicons. This significantly improves the sensitivity and specificity of PONT, making them comparable with or even superior to the sensitivity and specificity of polymerase chain reaction, considered as the “gold standard” of DNA diagnostics. The review considers modern approaches to the coupling of CRISPR/Cas detection using Cas9, Cas12a, Cas12b, Cas13a, Cas14, and Cas3 nucleases to various methods of nucleic acid isothermal amplification, with an emphasis on works in which sensitivity at the level of single molecules (attomolar and subattomolar concentrations of the target) is achieved. The properties of CRISPR/Cas nucleases used for targeted DNA diagnostics and the features of methods of nucleic acid isothermal amplification are briefly considered in the context of the development of diagnostic biosensing platforms. Special attention is paid to the most promising directions for the development of DNA diagnostics using CRISPR/Cas nuclease.