I own a 2010  toyota corolla S. we all know  to look for the general  maintenance stuff.  For me  my Car has 116K love it never put a red cent in her for any major repair.
I am having my own small issue i will  figure out. with the battery.
Fluids, you can always inspect the fluids.  Ask when the last changes were.  
Brake pads and drum brake when was the last replacements. Especially the  Disks and rotors.
Tires, look for wear & tear, and dry rot., Steal or alloy?  When was the last alignment, front  & rear ?
Any repairs done?? papers on it, if in the dealer or personal?  Car fax only works if he brought to a Dealer.
A magnet to check for Body repair. any signs of damages.
Interior Lights come on, radio did it ever turn off due to heat blowing on it? 
Ac is the O2   ever changed and when, is it original is it cold, the heat should pop on in 3 minutes time. 
Ac cabin filter,  air element filter, transmission filter, Fuel pump filter?  When were these ever changed, what gas stations you used?
City and or Highway, shortest trip, longest trip. 30 mpg minimum on highway 300 mi. trips
The list can go on and on best advice is make an appointment with your service shop,  if allowed if not find a trusted machanic and  get it inspected, that is a 1980's old school thing we used to do  have a pro look it over.
All in all you never know and on a optimistic point.  the Corolla is a great fuel economy and run about never let me down of the 8 years I have her and the three years in our lease, we bought the lease out due to my car  giving out. best  thing I ever did. but miss my other car Lol.
good luck and Please do come back with a follow up.
FISH