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$-$0935. We develop a refined Bayesian framework for determining how red noise strength scales with pulsar spin frequency ($ u$) and spin-down frequency ($dot{ u}$), 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 $ u^{a} |dot{ u}|^{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.
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