Most of the Southern Hemisphere's population lives nowhere near consistent snow. Australia's, New Zealand's and South Africa's major cities all sit at low elevation in relatively temperate latitudes, which means genuine sea-level snow is rare enough that when it happens, it makes the news for days. Here's when it's actually occurred.
Australia
Snow falling all the way to the coast on the Australian mainland is genuinely uncommon, but it has happened, and the cases tend to follow the same pattern: an unusually deep, cold low pressure system tracking close to the southeast coast, dragging polar air far enough north to override the normal thermal buffer the ocean provides. The most frequently cited near-sea-level events have affected parts of Tasmania and the far south coast of Victoria and New South Wales, where snow has occasionally settled at very low elevations, in some cases down to just above sea level, during the most intense cold outbreaks. These events are rare enough that a specific coastal town seeing settling snow is treated as a genuinely newsworthy occurrence rather than an annual expectation, and it tends to require both extreme cold aloft and precipitation timing to line up simultaneously, a combination that doesn't happen often even in a typical cold winter.
Tasmania sees this more often than the mainland, owing to its more southerly latitude and closer exposure to Southern Ocean systems. Hobart and other low-lying Tasmanian towns have recorded brief settling snow at or very close to sea level during the coldest outbreaks, something that happens with more regularity than anywhere on the Australian mainland, though still only during the most severe cold snaps each decade.
New Zealand
New Zealand's South Island sees sea-level snow with more regularity than mainland Australia, largely because it sits considerably further south and has a mountainous spine that channels cold southerly outbreaks directly onto low-lying areas. Christchurch and Dunedin, both close to sea level, have recorded settling snow during significant polar outbreaks, and the South Island's southern and eastern coastal strips are the most likely parts of the country outside the alpine regions to see snow actually reach the ground at low elevation. The North Island sees it far less often, generally requiring an exceptionally strong and rare southerly outbreak to bring snow down to sea level even in its southernmost regions.
Why it's rare, and what has to line up
Sea level snow in these regions comes down to the same basic constraint in every case: the surrounding ocean at these latitudes is simply too warm, most of the time, to let snow survive the trip from cloud base to the ground without melting into rain along the way. It takes a genuinely unusual airmass, one that's not just cold at altitude but cold enough all the way down to the surface, arriving at the same time as active precipitation, for snow to actually settle rather than fall as sleet or cold rain and melt on contact.
This is the same 850hPa-temperature logic that governs snow level in the Australian Alps, just taken to its most extreme conclusion. Where a typical alpine snow event might need an 850hPa temperature a few degrees either side of zero to bring the snow line down to resort elevation, a genuine sea-level event needs that same kind of cold air mass to persist essentially unmodified from several kilometres up all the way down to near ground level, over the ocean, in a part of the world where the sea surface temperature is actively working against it the whole way.