A phase 1 search is a painstaking, thorough attempt, in and around a starship, to find a person who is presumably injured and unable to respond.
However, since the basic assumption of the search is that the subject wants to be found, a capable person knowledgeable in the layout of the location searched and determined to evade the search can do so.
In 2267, a phase 1 search was initiated on the USS Enterprise in an attempt to find Lieutenant Commander Ben Finney, who was presumed to have been in ion pod, ejected to save the ship. Finney had actually faked his disappearance in an attempt to frame Captain Kirk for criminal neglect. (TOS: "Court Martial")
In 2369, Captain Jean-Luc Picard ordered a phase 1 search when Data detected a warp engine breach from the Shuttlecraft Justman during Dr. Beverly Crusher's attempt to prove Dr. Reyga's metaphasic shield worked. However, the Justman emerged from the corona of Vaytan before the search could be implemented. (TNG: " Suspicions")