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!