If the vehicle will not crank over, the most common problem is the starter, which tend to fail at about 100,00-125,000 miles. Sometimes it is only the starter solenoid contacts, but often the complete starter (including solenoid) needs to be replaced.
Resources for the Repair and Maintenance of your 1992 Toyota Celica All-Trac
1992 Toyota Celica All-Trac
1992 Toyota Celica All-Trac Problems
The power steering pump and power steering hoses tend to develop leaks and require replacement.
If the vehicle develops an annoying, clunking sound when driving over bumps it is usually the struts and strut bearing plates. These may need replacing at 100,000-125,000 miles or sooner. If the noise is a more of a squeak or groan, it can be usually be eliminated by spraying a light lubricant on the suspension bushings 1-2 times per year.
Usually by 150,000 miles the AC system will need some attention, especially in climates where it is used often. Our technicians tell us that it is best to service the AC system every 2-3 years. This keeps the moisture in the system at a minimum which extends the life of the components.
Our technicians report that by 100,000 miles, the ignition key tends to wear out and should be replaced or it may not work as well in the door locks or ignition lock cylinder.