Several steps of an investigation of single top-quark production at the Large Hadron Collider (LHC) are discussed. A number of general problems associated with basic steps of this analysis are discussed. The investigations are performed for the CMS (Compact Muon Solenoid detector) Collaboration, and detailed results will be presented in CMS publications. Three event generators used by the CMS Collaboration in simulating single top-quark production (SingleTop, MC@NLO, and a generator based on the MadGraph package) are compared in the present study. Schemes for optimizing the observation of a single top quark on the basis of the neural-network method are presented. The first steps in searches for possible anomalous vector couplings in the Wtb vertex that manifest themselves in processes of single top-quark production are described.