  The low-frequency linearly-polarised radio source population is largely
unexplored. However, a renaissance in low-frequency polarimetry has been
enabled by pathfinder and precursor instruments for the Square Kilometre Array.
In this second paper from the POlarised GaLactic and Extragalactic All-Sky
Murchison Widefield Array (MWA) Survey -- the POlarised GLEAM Survey, or POGS
-- we present the results from our all-sky MWA Phase I Faraday Rotation Measure
survey. Our survey covers nearly the entire Southern sky in the Declination
range $-82^{\circ}$ to $+30^{\circ}$ at a resolution between around three and
seven arcminutes (depending on Declination) using data in the frequency range
169$-$231 MHz. We have performed two targeted searches: the first covering
25,489 square degrees of sky, searching for extragalactic polarised sources;
the second covering the entire sky South of Declination $+30^{\circ}$,
searching for known pulsars. We detect a total of 517 sources with 200 MHz
linearly-polarised flux densities between 9.9 mJy and 1.7 Jy, of which 33 are
known radio pulsars. All sources in our catalogues have Faraday rotation
measures in the range $-328.07$ rad m$^{-2}$ to $+279.62$ rad m$^{-2}$. The
Faraday rotation measures are broadly consistent with results from
higher-frequency surveys, but with typically more than an order of magnitude
improvement in the precision, highlighting the power of low-frequency
polarisation surveys to accurately study Galactic and extragalactic magnetic
fields. We discuss the properties of our extragalactic and known-pulsar source
population, how the sky distribution relates to Galactic features, and identify
a handful of new pulsar candidates among our nominally extragalactic source
population.
