“Parable of the Sower” by Octavia Butler is an excellent read. This book is about a dystopian future United States. The cities are walled off and mankind is plagued with disease and mental instability. Lauren Olamina is an 18-year-old teen with the abilities to feel the pain of others. She chooses to leave her home and this is a story of her journey.
“A Gift Upon the Shore” is another good novel. It was written in 1990 by M. K Wren and relies strongly on a fundamentalist philosophy. The story is set in a world of post nuclear holocaust. There are two female protagonists who struggle with survival through the diseases and nuclear fall-out.
Another dystopian fantasy is “The Slynx” by Tatyana Tolstava. This novel was written in 2000 by the Russian author and was translated into English in 2003. The story begins 200 years after “the blast” and the society which evolves is evil and dangerous.
A novel which has new life, due to a recent movie adaptation, is “The City of Ember” by Jeanne DuPrau. Ms. Duprau’s novel was published in 2003 and has the same post-apocalyptic end-of-the-world theme. The society that we know had been destroyed centuries before this novel begins but it is still a great read. The protagonists are children looking for a solution to the lack of energy. Unless they can find a solution Ember will be as dark as the rest of the world.
There are many others including the classics like “On The Beach.” by Nevil Shute “Alas, Babylon,” by Pat Frank, ”Level 7,” by Mordecai Roshwald and “Earth Abides” by George R. Stewart. Hope this helps.
The Lord of Flies is a great one, as are Huckleberry Finn, Frankenstein, and The Red Badge of Courage. I've also always loved the story "Rikki Tiki Tavi" which has a very strong theme of survival. Night and The Poisonwood Bible are two others where survival is a frontrunning theme. There is a series entitled Among the Hidden where it is illegal to have more than a certain number of children. The ones born outside that number have to hide and live in secrecy or risk being killed. They are pretty good books, and survival is tantamount to these kids. "The Most Dangerous Game" is another story where survival is foremost on the main character's mind as he is hunted by a famous hunter who wants to put his head on a stake.
There are tons of great survival stories out there! I"ve included a link here at enotes that gives a long list of books and stories which deal in some way with the theme of survival. Good luck and good reading!
I can think of several "lone person survives awful odds" stories. One book that comes to mind is "Life of Pi" by Yann Martel. It is about a little boy from India, whose family is travelling to Canada on a boat. His family owns a zoo, so there are animals on the boat. The boat sinks, and Pi ends up on a lifeboat and has to survive; the twist is that there is a Bengal tiger on the lifeboat with him. He survives, at sea, with the tiger, for an unprecendented amount of time. Granted, all of society is not facing utter destruction, but, this little boy is.
Other classic survival stories are "Hatchet" by Gary Paulsen, where a boy survives a plane crash in the wilderness and must learn to survive. "Into Thin Air" by John Krakaur is about the true account of survivors of a plane crash on the frozen top of a mountain.
For other really great survival stories, where utter destruction is involved, look more to the science fiction genre. In many futuristic science fiction plotlines, mankind is on the brink of extinction, or is facing odds against alien invaders. In the "Ender's Game" series by Orson Scott Card, mankind faces destruction against a race of aliens called "Buggers," and one brilliant child, Ender, is being trained to save mankind.
I hope that these suggestions helped; I am sure that other editors will chip in with more suggestions also. Good luck!