Brush-tailed Possums have a wide distribution over Australia, being more common in urban areas of the southern and eastern states, but rarer in arid regions (Strahan, 1995).
They tend to be solitary, although how many are found in an urban area and how far they roam is tied to availability of suitable food and places to rest (Bird, 1997). Possums are an example of one of the few Australian animals that have readily adapted to environmental changes wrought by Europeans over the past 200 years (others include Australian Magpies and Eastern Brown Snakes).

HOW DO YOU IDENTIFY THE OFFENDER?
Before you set in action, any management methods, a correct identification of the pest responsible for the damage is necessary. Possum foraging is very distinctive. Possums are messy eaters and the ground will be cluttered with half eaten leaves, flowers, or fruit. The leaf stalk and midrib are usually left behind in broadleaf species of plants and you will find small branches may be broken near the top of the tree. When manoeuvring from one area to another, the possums may leave trail damages, especially through grassed areas. You may find visible claw marks on the trees and fence posts. Possum droppings are usually 2.5cm long,(thicker than a pencil) and covered in a slimy film when fresh. Possums are highly territorial animals, fights can break out and sometimes be heard or you may find fur pieces that are left behind.
Possums feast on an assortment of native and introduced vegetation. By continually grazing on particular trees they can eventually destroy them consequently transforming the structure of the forests. Possums will plunder produce from orchard trees, damage shelter belts and feed on crops, flower and vegetable gardens. Rose buds are a great delicacy to possums.

Problems (Coombe, 1997)

Research in recent years has shown that possums in urban areas will utilise artificial den sites (roof spaces, chimneys, under or in buildings) more than they will tree hollows when both are available (Statham and Statham, 1997).
The problems caused by possums in urban areas are many and varied. Following are some of the more obvious:

Noise. Sounds of possums bounding about in the ceiling space and/or vocalising with other possums. Often a householder has mistaken rats in the roof for possums.
Sleeping inside. Sometimes the sounds of a possum in a chimney, cavity wall or ventilator have led the house occupant to think the animal is trapped, when it is not.
Household damage. A traumatised possum that either falls down a chimney or inadvertently finds its way inside the house can cause considerable damage when attempting to escape.
Faeces contamination. Amongst the first evidence of a possum in the garden are the droppings on a path or driveway, or washed from the gutters into the rainwater tank.
Stains. Characteristic urine stains on ceilings or walls inside.
Odours. When a Possum dies in the roof there is an overpowering stench. To help eradicate the odour, remove the dead animal and with sprinkle Orris Root powder( available from healthfood shops) or place a small piece of rag soaked in Nil Odour in a container and place over area where the creature laid.
Disturbance. Possums are sometimes blamed for upsetting the neighbourhood dogs, when a cat passing through the yard is as likely to be the culprit.
Damage to gardens. Possums have developed a taste for many types of garden shrubs, vines, fruit trees, vegetables and grasses.
Neighbourhood disputes. One property owner may be happy to have possums in their yard while across the fence another person may be just as keen not to see, hear, smell or otherwise know of their presence.

Management Options (Coombe, 1997)

Traditionally, offending possums have been relocated "down the road" in the nearest (furthermost?) patch of bush. However, research has again demonstrated that the majority (88%) of such relocated animals do not survive more than a week (Pietsch, 1995). The "fact" that relocated possums will find their way home has been disproved by Statham and Statham, (1997) who discovered that none of the studied animals returned to the point of capture.
Many techniques have been tried to minimise possum problems. Following are some of them:

Trees near buildings. Prune overhanging branches that provide access to the roof. If that is not feasible then fit metal collars around the trunk of any offending trees.
Destroying scent. Brush-tail Possums communicate by scent as well as sound, so remove their scent at points of entry into a building by washing with a bleach solution.
Deterring possums from roof spaces. Some chemicals have been suggested that may deter possums from a roof space. They include cloves of garlic, camphor or naphthalene (but not both together), mosquito coils or POSS-OFF (a natural Possum deterrent). A bright light on during the day may also help.
Inside house. If a possum is "trapped" inside a chimney it can be assisted to escape by dangling a rope down the cavity from the outside. If the animal is inside a room, open exterior door(s) and windows and gently coax it outside.
Trapping. If all else fails, catch the possum with a wire mesh possum trap set up in the roof space. Some pet shops, councils and wildlife groups have them for hire. The trap can be baited with chopped fruit, bread and honey or even a rag sprinkled with aniseed essence. Once removed, all points of entry must be found and sealed to prevent re-entry. Or ask a specialist contractor to do it for you (check the Yellow Pages for details).
Nesting boxes. Since research has shown that relocating a possum does not solve the problem for you or the animal, providing an alternative home (a wooden nest box) is the better way to go. Numerous creatures will willingly adopt man-made types of boxes and will rapidly adjust to their new habitat.
They can be bought from similar sources as possum cages.
Investigations show possums which have been moved from their home are often killed by other possums protecting their own territory. The animal continuously being attacked, can cause a slow death, due to infections or blindness caused by injury or by starvation.
If you have possums in your roof, wait until they go out to feed at night and then possum-proof the area that they are getting in.
Protecting plants. If it is practical, possums can be physically excluded by fencing. An unsupported 'floppy-topped' fence can be added to an existing fence. It is more likely to prevent possum intrusions than a normal fence. Large fruit trees can have a metal collar fitted. Smaller plants or vegetables can be covered with plastic bird netting. Various chemical deterrents may include quassia chips,garlic spray, D-Ter, POSS-OFF (a natural Possum deterrent) or even spreading blood and bone fertiliser around the base of plants. 
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References.

Coombe, G.N. (1997). Problems Caused by Common Brushtail Possums and Management Options. In: Proceedings of 'Seminar and Workshop on the Management of the Common Brushtail Possum'. Fauna Management Coordinating Committee, Adelaide May 29, 1996.
Bird, P.L. (1997).A Review of the Ecology of the Common Brushtail Possum Trichosurus vulpecula in South-Eastern Australia. In: Proceedings of 'Seminar and Wokshop on the Management of the Common Brushtail Possum'. Fauna Management Coordinating Committee, Adelaide May 29, 1996.
Pietsch, R.S. (1995). The Fate of Urban Common Brushtail Possums Translocated to Sclerophyll Forest. In: Reintroduction Biology of Australian and New Zealand Fauna. Surrey Beatty & Sons.
Strahan, R. (ed)(1995). The Mammals of Australia. Reed Books.
Statham, M. & Statham, H.L. (1997). Movements and Habits of Brushtail Possums (Trichosaurus vulpecula Kerr) in an Urban Area. Wildlife Research 24: pp 715 - 726.