In this paper we report the current status of a new research program. The primary goal of the "Vanishing and Appearing Sources during a Century of Observations" project is to search for vanishing and appearing sources using existing survey data to find examples of exceptional astrophysical transients. The implications of finding such objects extend from traditional astrophysics fields to the more exotic searches for evidence of technologically advanced civilizations. In this first paper we present new, deeper observations of the tentative candidate discovered by Villarroel et al. (2016AJ....152...76V). We then perform the first searches for vanishing objects throughout the sky by comparing 600 million objects from the US Naval Observatory Catalogue (USNO) B1.0 (Cat. I/284) down to a limiting magnitude of ~20-21 with the recent Pan-STARRS Data Release-1 (DR1, Cat. II/349) with a limiting magnitude of ~23.4. We find about 150000 preliminary candidates that do not have any Pan-STARRS counterpart within a 30" radius. We show that these objects are redder and have larger proper motions than typical USNO objects. We visually examine the images for a subset of about 24000 candidates, superseding the 2016 study with a sample 10 times larger. We find about 100 point sources visible in only one epoch in the red band of the USNO, which may be of interest in searches for strong M-dwarf flares, high-redshift supernovae, or other categories of unidentified red transients.