  While pulsars possess exceptional rotational stability, large scale timing
studies have revealed at least two distinct types of irregularities in their
rotation: red timing noise and glitches. Using modern Bayesian techniques, we
investigated the timing noise properties of 300 bright southern-sky radio
pulsars that have been observed over 1.0-4.8 years by the upgraded Molonglo
Observatory Synthesis Telescope (MOST). We reanalysed the spin and spin-down
changes associated with nine previously reported pulsar glitches, report the
discovery of three new glitches and four unusual glitch-like events in the
rotational evolution of PSR J1825$-$0935. We develop a refined Bayesian
framework for determining how red noise strength scales with pulsar spin
frequency ($\nu$) and spin-down frequency ($\dot{\nu}$), which we apply to a
sample of 280 non-recycled pulsars. With this new method and a simple power-law
scaling relation, we show that red noise strength scales across the
non-recycled pulsar population as $\nu^{a} |\dot{\nu}|^{b}$, where $a =
-0.84^{+0.47}_{-0.49}$ and $b = 0.97^{+0.16}_{-0.19}$. This method can be
easily adapted to utilise more complex, astrophysically motivated red noise
models. Lastly, we highlight our timing of the double neutron star PSR
J0737$-$3039, and the rediscovery of a bright radio pulsar originally found
during the first Molonglo pulsar surveys with an incorrectly catalogued
position.
