The species of Geastrum are known colloquially as earthstars. Each mature fruitbody consists of a puffball-like spore sac atop a star-like base. Some species are easy to identify visually, others are not. Before you started looking at microscopic details, some of the questions you'd ask when trying to identify a Geastrum are:
Does the spore sac sit directly on the base or is it held up by a stem (and if so, what is its size)?
Are the arms of the base hygroscopic (in other words, do they fold inwards in dry conditions and open out in moist conditions)?
Does soil/debris adhere tightly to the undersides of the basal arms or are they free of soil/debris (or at least where any soil/debris is easily removed by light brushing)?
What is the nature of the apical mouth, through which the spores puff out? Two examples are a (1) conical, strongly pleated mouth (as shown here, http://www.cpbr.gov.au/fungi/images-captions/geastrum-sp-ostiole-0144.html) and (2) a fibrillose mouth (http://www.cpbr.gov.au/fungi/images-captions/geastrum-sp-ostiole-0143.html).
You'd also note colour, texture and size.
Geastrum sp. is listed in the following regions:
Canberra & Southern Tablelands | Southern Highlands | South Coast | Central West NSW | Loddon Mallee
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Maps
ANBG Bicentennial Park Black Mountain Boro Bruce Ridge Bullen Range Cooma North Ridge Reserve Fadden Hills Pond Federal Golf Course Ginninderry Conservation Corridor Goulburn Mulwaree Council Hall Horse Paddocks Hughes Garran Woodland Hughes Grassy Woodland Isaacs Ridge Isaacs Ridge and Nearby Lake Ginninderra Lower Cotter Catchment Mcleods Creek Res (Gundaroo) Molonglo Gorge Molonglo River Reserve Mount Ainslie Mount Majura Mulligans Flat Namadgi National Park National Arboretum Forests Pine Island to Point Hut Red Hill Nature Reserve Red Hill to Yarralumla Creek The Pinnacle Tidbinbilla Nature Reserve Uriarra Recreation Reserve Wanniassa Hill Wright Park and Old Sydney Road ReservePlaces
Queanbeyan West, NSW Yarralumla, ACT