Probably the most important point on your checklist, the engine is the heart of the car. Although engines can put up with a lot of wear and tear its maintenance is key. The easiest things to spot are leaks. Lots of different fluids run in and around the engine, but a properly maintained car shouldn't spring a leak. Check underneath the car for any signs of oil on the tarmac, or any sludge on the bottom of the car itself. Be sure to also open the hood and check for leaks around the engine, where you might spot either oil leaks (usually a brownish color), coolant, or gearbox fluid. If you see any of this leaking, either ask for money off the price of the car, or for the seller to fix it as part of the deal.
Checking the oil should also be standard procedure when inspecting a used car. Make sure the dipstick reading is accurate and that the oil is the correct consistency and color. While you're there, look for any signs of dirt covering any connectors, as this could be a giveaway that the vehicle has been poorly maintained or serviced.
3. Gearbox And Clutch
One of the primary reasons it's important to take a used car for a test drive is that it's the only way to check the condition of the gearbox and clutch. Listen for any unusual sounds when changing gear - or accelerating and decelerating if your car has automatic transmission - because a healthy gearbox should engage the gears smoothly and quietly. If your car has a manual gearbox, make sure the clutch 'biting point' is around the middle of the pedal's travel - you shouldn't need to push your foot to the floor.
4. Wheels And Tires
Be sure to check all four wheels, plus the spare wheel if the car has one. The odd graze from a curbstone isn't a major issue, but dents or bends in the wheel rim should be flagged. Tires aren't cheap to repair, so make sure that they are in decent enough condition. They should be free from cuts, splits, and bulges. Consistent under-or-over-inflation will also wear the tires down in the middle or on the edges, so check for signs of this as well.
The condition of a used car's interior can help determine whether the mileage is genuine. If a car is showing less than 20,000 miles on the dashboard then the interior should be like new, so if it isn't, perhaps the seller is being dishonest. Check for wear and tear in the upholstery and any unusual damage to the dashboard. Also, be sure to test the equipment - lights, windows, air-con, etc. and ask for a discount if it's in need of repair.