I've had my Cobalt since 2007 and the only major problem I've had was my fault (dinged the oil pan). Replaced the engine for less than $1,000 and was back on the road again.
Gas mileage is unparalleled. I usually get 35mpg+, sometimes up to 40mpg on long trips. I drive the hell out of this car and it never skips a beat.
My largest complaints are aesthetic: the model I have (2005 sedan, energy blue) looks cheapish. The interior seats aren't the most comfortable, but it's really only noticeable from the passenger side. >)
As this car ages, a few benign electrical problems have popped up, but nothing serious.
Great value if you're looking for a cheap investment and long-term savings.
Gas mileage! I have a long commute and this little car gets up and goes without sucking up gas.
I frequently take long country drives on very little fuel.
It's reasonably fast for it's small-sized engine.
The top-loading oil filter is pretty handy and unique. The car is low, so you'd need a jack to get under there, but that problem is solved by replacing the filter from the top of the engine.
Interior seats need more padding (at least in my 2005).
Could be more stylish.
I don't like the auto-arming alarm because I don't have a fob, so it beeps whenever I get in. Needs to have a disable option.
Doors covers are unreasonably hard to get into if you need to replace a speaker or switch, but from what I understand this is a Chevy problem and not a Cobalt problem.
Not sure if this is true with all Cobalts, but some of the wiring for my headlight assembly just hangs out. Because of this, one of blinkers/parking lights has a short in it. The wires need to be clamped down to the side of the chassis and not allowed to rub against different things. Not a killer, but definitely a pain.