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We present a search for merging compact binary gravitational-wave sources that produce a signal appearing solely or primarily in a single detector. Past analyses have heavily relied on coincidence between multiple detectors to reduce non-astrophysical background. However, for $\sim40\%$ of the total time of the 2015-2017 LIGO-Virgo observing runs only a single detector was operating. We discuss the difficulties in assigning significance and calculating the probability of astrophysical origin for candidates observed primarily by a single detector, and suggest a straightforward resolution using a noise model designed to provide a conservative assessment given the observed data. We also describe a procedure to assess candidates observed in a single detector when multiple detectors are observing. We apply these methods to search for binary black hole (BBH) and binary neutron star (BNS) mergers in the open LIGO data spanning 2015-2017. The most promising candidate from our search is 170817+03:02:46UTC (probability of astrophysical origin $p_{\rm astro} \sim 0.4$): if astrophysical, this is consistent with a BBH merger with primary mass $67_{-15}^{+21}\,M_{\odot}$, suggestive of a hierarchical merger origin. We also apply our method to the analysis of GW190425 and find $p_{\rm astro} \sim 0.5$, though this value is highly dependent on assumptions about the noise and signal models.