So I've put a lot of thought into this and I fairly confident in my prediction of which areas of the world would survive.
North America:
there would be there main survival points in North America. The Seattle, Portland, Toronto, area due to it's proximity to the Rocky Mountains, and a defensible access point to the ocean. Hover Damn, and the Mojave (maybe Vegas but probably not). The Mojave desert would be very effective in eliminating trolls, as well as the water supply of lake Mead, the electricity of the dam itself, and the ability to set up hydroponic farms. Not to mention Nevada is home to a very large solar power plant that with some skill could be reactivated. Richmond, and Norfolk Virginia, as well as the nearby sections of the Appalachians. Another scenario of mountains for initial survival and then a reclaimed defensible port. Also while I don't think any other major cities would be reclaimed however Alaska and most of northern Canada would have small groups of survivors.
South America:
The western edge of South America is at very high elevations and the southern end is close to Antarctica. While Central America, and the Amazon rain forest, as well as the previous heavily populated areas of Brazil would be highly dangerous, Chile, Argentina, and the western part of Peru would be mostly fine.
Africa and Australia:
While theses places are different in people there survival would face similar difficulties. While the desert and sparse grasslands would offer protection from trolls and beasts, the main issues is that in both of these places the most densely populated areas are the ones with ready access to water. Thus leading to a large portion of the water supply becoming non accessible to do troll nests and beast hunting grounds. While it is possible for the survivors to all band together and retake access to water it is also likely for these water supplies to fall into the hands of the ill willed, leading to a very Mad Maxian scenario.
New Zealand:
While Australia is, as we in the industry like to call, ducked, New Zealand would be relatively fine. It isn't densely populated, it has plenty of mountains and cold enough weather for survival.
South East Asia and India:
While I list these areas the only survivable area in this region is Nepal/the Himalayas. India has dense population surrounding water, and south east Asia just doesn't have the climate. To warm and lots of rain storms that would block the sun.
North East Asia:
While the main cities of China would be gone, and Taiwan would be reduced to at most a few small survivor groups. North east Asia is where it gets interesting and could lead to a similar scenario to SSSS. North Korea would be almost surly destroyed but the DMZ would serve as a defensive line for South Korea. Not to mention while Soul is densely populated many Koreans who live there have plans in place to leave the city in the event of an emergency due to the cities proximity to the DMZ. Japan would be mostly fine, and I'm not saying that as a weab, it actually has good mountain ranges and climate, not to mention large amounts of rural land. Also Mongolia would be almost 100% fine. The Mongolians have maintained there traditions and ways to this very day, they are still amazing mounted archers and live mostly nomadic ally.
I understand there are a lot of areas that I didn't list. I simply do not know enough or I consider a near flat 50-50 chance of survival. So if I didn't list it here I could be swayed either way. Also while I'm happy to debate this I don't even consider this fact. I'm using estimations and rough knowledge. Survival depends on people working together. If people cooperated the entire world could be retaken or if they don't it could be absolute chaos. So in general if you think a regions people would put aside their differences and cooperate and you want to write about that area go ahead, also if you think the regions I listed are too set in their ways to cooperate and you want that area to be chaos go right ahead.