Trichoderma 'green fluffy'

This is a mouldy growth, initially white but soon greyish green, though the margins often remain white. In overall appearance, there is some resemblance to green-stained cotton-wool. Separate colonies may merge to cover many square centimetres

Green fluffy Trichoderma's are very common on soil  - also on rotten wood and I've seen them also on some fungi (e.g. the bases of old Scleroderma or Pisolithus fruit bodies).

Trichoderma viride is possibly the best known species (and earliest name, dating from 1794) and one you are very likely to come across in any web search. There are more species and they cannot be identified from photographs alone and so are lumped together under the epithet 'green fluffy'. The two papers listed below tell you something of the variety in Australasia. Of local interest is Jaklitsch et al's report of Trichoderma gamsii isolated from xylem of Eucalyptus nitens from the Australian National Botanic Gardens.  

The 'green fluffy' Trichoderma's are NOT the greyish green moulds that you find on bread or fruit.

Look-alikes

Some other genera may also produce greenish mouldy growths on soil or rotting wood. However, if you see a good sized, white-fringed, grey-green, 'cotton-wool' mould on soil or well-rotted wood, assume Trichoderma.  

References

Braithwaite, M. et al, (2017), Trichoderma down under: species diversity and occurrence of Trichoderma in New Zealand, Australasian Plant Pathology, 46, 11–30.

Jaklitsch, W.M. et al, (2006), Hypocrea rufa/Trichoderma viride: a reassessment, and description of five closely related species with and without warted conidia, Studies in Mycology, 55, 135–177