A common weather pattern in Melbourne is one that plays out over a week to 10 days, with the passage of the high/low pressure systems. A front will arrive with rain/grimness/storms, and turn the winds to the south/southwest. That usually means anything from cloudy to quite unpleasant across Vic. The front(s) will clear, but leave us in a spell of southwesterly winds, with a high pressure system based somewhere near the Great Australian Bight. This often lasts about 3-6 days or so, with onshore breezes and clouds that are initially quite thick, thickest towards the coast, perhaps with showers, gradually reducing as pressure builds (the high system sliding eastwards and getting closer to Melbourne). If we're lucky, the high will pass over Melbourne, giving sunnier days for a day or two, before the winds shift into the north/northwest after the high. That time is our best chance for clear skies, with the winds coming off the continent, but it only lasts until the arrival of the next set of fronts, starting the pattern over again.
In summer, there's quite a lot of chance of sun, as the lows are usually further out to sea, and the sun is stronger so it can 'burn off' stratiform cloud that comes in from the coast on the southwesterly/southerly winds. That means northern Vic has bettern chances of clear skies, but Melbourne can do OK in summer. In winter, the sun isn't so effective at doing this, and the lows pass a bit closer, all adding up to much longer spells of cloudiness. You wait patiently (!) for the wind to turn into the north after a high pressure arrives/passes by, then that only lasts for a day or so ahead of the next rain band. Sod's Law of course will put that window of opportunity at Full Moon as pgc hunter says (or in the case of this week, with a big fat evening gibbous Moon), but when spring returns and a stronger Sun, and the lows not passing so close, we should have more chances of clearer skies.
I'm not so sure about evening versus morning patterns, but you have more chance of good evening conditions at least in part because the air temperature will be further from the dew point at dusk relative to later in the night (which means risk of dew as well as mist and poor transparency if relative humidity gets too high near the surface).
I'm reckoning on about one or two clear nights per New Moon period at the moment, but in spring/summer/autumn it was more like five or six at least. Of course I'll end up having social plans made for those nights well in advance...
For good satellite images, I've been using:
http://realtime2.bsch.au.com/vis_sat.html
You get much better satellite images than on the BoM and can zoom in on Vic. On the visible satellite images, you can get a nice 'feel' for whether coulds are melting back to the coast or not, and other changes of weather by setting up animations of the hourly images. You can use IR images for overnight, but beware that low (warm) clouds often do not show up well on night-time IR imagery.