  White dwarfs (WDs) are powerful tools to study the evolutionary history of
stars and binaries in the Galaxy. But do we understand their multiplicity from
a theoretical point of view? This can be tested by a comparison with the sample
of WDs within 20 pc, which is minimally affected by selection biases. From the
literature, we compile the available information of the local WD sample with a
particular emphasis on their multiplicity, and compare this to synthetic models
of WD formation in single stars and binaries. As part of our population
synthesis approach, we also study the effect of different assumptions
concerning the star formation history, binary evolution, and the initial
distributions of binary parameters. We find that the observed space densities
of single and binary WDs are well reproduced by the models. The space densities
of the most common WD systems (single WDs and unresolved WD-MS binaries) are
consistent within a factor two with the observed value. We find a discrepancy
only for the space density of resolved double WDs. We exclude that
observational selection effects, fast stellar winds, or dynamical interactions
with other objects in the Milky Way explain this discrepancy. We find that
either the initial mass ratio distribution in the Solar neighbourhood is biased
towards low mass-ratios, or more than ten resolved DWDs have been missed
observationally in the 20 pc sample. Furthermore, we show that the low binary
fraction of WD systems (~25%) compared to Solartype MS-MS binaries (~50%) is
consistent with theory, and is mainly caused by mergers in binary systems, and
to a lesser degree by WDs hiding in the glare of their companion stars. Lastly,
Gaia will dramatically increase the size of the volume-limited WD sample,
detecting the coolest and oldest WDs out to 50 pc. We provide a detailed
estimate of the number of single and binary WDs in the Gaia sample.
