The domestication of goats has resulted in extinction of many plants species and there is academic debate as to whether they or fire are more to blame for the general lack of forests in the Mediterranean. There is no debate that they prevent the healthy regeneration of native bush in New Zealand. Much like petroleum or booze their great power appears to offer the easy way. But like anything more powerful than us their effects are by definition difficult or impossible to control. Unless used with the greatest of prudence such addictive power actually makes life harder, even impossible in the end.
Awareness of their indiscriminate destructiveness has grown in recent years and fashion and legislation are slowly catching up. It has recently become illegal for your neighbors goats to be on your land even with your permission in Wellington (they must be confined to the owners property (assuming of course this wasn't careless wording, quite possibly they meant to say "goats must be confined to properties where they have permission to be")) but if dobbing in your neighbor to the Council is not your style I suggest resorting to the miraculous divaricating Coprosmas.
There are other divaricating native genera that also evolved to resist grazing by the extinct Moa, but I have not noticed them mentioned as survivors in studies of effects of wild goat infestation. The fashionable divaricating Meuhlenbekia genus for instance does not seem to be nearly as resistant to introduced herbivores like goats).
Divaricating Coprosma are desirable because they are not actually poisonous to goats like many goat resistant plants such as Rhododendron or Avocado. It is the physical structure of the plants that prevents them from eating them, not taste or toxicity. This avoids psychological damage to the owner of the goats you are trying to keep out, and possibly even legal liability from deliberately planting a poisonous hedge to kill trespassing goats (I don't know if prosecution for this has ever happened but suppose it could).
In addition they are a dense, sturdy, wide plant that may be difficult or impossible for goats to penetrate, climb over, ring-bark or trample as they might with most other plants unpalatable to goats. I have seen divaricating Coprosmas largely untouched and undamaged in sheep and cattle farms, though they are found in relatively inaccessible spots, presumably where the young plants have had a chance to grow without being trampled.
In my experience goats do not eat small young divaricating Coprosma though tethered goats are likely to trample them to death with their erosion causing hooves, as they do many other plants unpalatable or toxic to them. It is likely goats tethered next to divaricating Coprosma with absolutely nothing else to eat would devote their lives to trying to eat it, whether they would succeed I don't have data.
Because of the tendancy of goats to trample plants to death these Coprosma will probably need fencing when young. I have a mesh fence 1.6 meters tall as a first defense with divaricating Coprosma behind them as second line of defense. Also this hopefully makes them out of sight and out of mind, they can still be quite intimidating and disturbing gawking mindlessly at your fruit and vegetables through a mesh fence even if they are really unable to do any physical harm. I have also planted my divaricating Coprosma close together, 1 meter apart at the most, to increase strength.
Divaricating Coprosmas grow in a wide variety of conditions, are fire retardant and most grow quickly to several meters tall. They are ornamental and berries of all Coprosmas are thought to be edible to humans, though not especially tasty in most cases. There is probably variation amongst the the divaricating Coprosmas in terms of efficacy against goats, I'm trying several.
Aside from personal observation of sheep and cattle farms in the Wellington area, and limited experience with tethered goats, I have based this 'thesis' on reports that divaricating Coprosma are found surviving in areas infested with wild goats:
ASSESSMENT OF ECOLOGICAL EFFECTS OF THE PROPOSED LONG GULLY WIND FARM, WELLINGTON May 2009 Report No. 2084
Christchurch District Plan Site of Ecological Significance Site Significance Statement Site name: Kinloch Site number: SES/A/11 Physical address of site: 184 Kinloch Road, Little River