CarFax is not supposed to be used to find out about accidents.
I was shopping for a car a couple months ago and ran a ton of CarFax reports. I can tell you that you can learn alot of valuable info from them. For instance:
-Check what kind of registration the car had. If it was a Commercial, Rental, or Fleet vehicle, there's a much higher chance that it wasn't taken care of as well as if somebody had bought it. There are probably more dings, stains in the carpet, etc. and probably wasn't maintained as well.
-Look at the mileage reports to see how it was driven. For instance, a 3 year old car that has around 30,000 miles might look like it has been driven less than average. BUT, I saw a few cars where they were driven around 2,000 miles total in a year, then driven 20,000 or 25,000 miles in the next 8 months. Definitely an irregular pattern of driving, and anytime you put a large amount of miles on a car in a short time it's not great for the car.
-If the car is at a dealership, usually you can tell how long it's been there by looking at the last date in the Title History. If you see one that's been at the dealership for over a couple months, chances are you can get a great deal if you haggle with them. Just tell em you've noticed it sitting there for a few months, and remind them they are losing money every day they don't sell it. Once they know you're not a moron they'll usually cut you a good deal to get it of their hands.