You won't find these on big dealership lots. Look where private sellers list older high-mileage vehicles:
I can give you a specific to search for in your area! buy a used car for under 1000 dollars
Do you have a in mind (sedan, truck, hatchback)? You won't find these on big dealership lots
Expect high miles (250k+) and cosmetic damage. buy a used car for under 1000 dollars
Ask the seller not to warm the car up. You want to see how it starts when cold.
If you have $1,000, try to buy the car for $700. You will need the other $300 for an oil change, a battery, or a belt.
You likely won't pay for a $150 mechanic inspection for a $900 car. Do these yourself: