I am pleased to report that I have never been caught driving a car like the one in Uncle Buck, where the title character leaves a little explosion upon each arrival he makes. That said, I haven't always driven a noiseless machine. I have definitely contributed my share of noise pollution to the roads I frequent simply by driving there. So what commonly heard car sounds require immediate attention, and which are not so serious? Let's take a look.
Thumping: is typically the result of a flat tire. If the flat occurred while the car was parked, then hopefully you'll notice it before you take off. Otherwise, you'll immediately feel and hear its thumping vibrating from the suspension. A flat tire at the front of your car will tend to drag the car towards the side of a flat. We really don't advice driving on a flat. It'll wreck your tire in no time. Without air in the tire, the sidewalls will get pinched between the road and the wheel rim. Even driving one measly kilometer with a flat is long enough to damage the tire, and if the tire detaches from the rim as flats often do, you could really damage your wheel as well.
Screeching: screeching with no other sound attached to it shouldn't alarm you if it's coming from inside your wheels, especially if you notice it more when you're slowing down, and when the weather exhibits damp conditions. Your brake pads are simply grating against the discs, but they still work. Adding new brake pads should do the trick for eliminating the sound.
Screeching that ends in a putter – if this sound is coming from under your hood, as it likely is, you won't want to take off for a road trip anytime soon. Get it checked out as soon as you can. That sound is the sign of a faulty belt, faulty tension, or a misaligned pulley.
Puffing – puffing that comes from under the hood, especially when your car is idle, is serious. It's a sign that an exhaust manifold is not working as it should, and that hot exhaust gases are being released into the air. It's illegal to drive a car in that condition, not to mention ultra dangerous for you, since it will expose you to carbon monoxide.
If ever you are uncertain, err on the side of caution and ask somebody—a mechanic, ideally, but a friend or family member who knows a thing or two about cars (expertise is not necessary) should be able to offer a helpful opinion as well.