"Speedball Tucker" by Jim Croce, and also "Rapid Roy".
Murdoc Niccals is a notoriously crazy driver, as seen in the Stylo and 19-2000 music videos. To be fair, he pulls some ridiculously cool stunts in the latter; wheelies, jumps, skids, missile launches etc.
"Road Man" by Smash Mouth, the tale of a roadie who Drives Like Crazy to get a touring band's sound equipment from Point A to Point B as fast as possible (and also just because he wants to be "king of the road"). His monomaniacal focus on speed eventually gets him killed when he doesn't notice a train until it's too late.
"Switzerland" by the comedy band Dead Cat Bounce is a song told from the perspective of one:
Such was the expression of the child as he bounced across my windscreen and off the other side. I got the strong impression for a second that he wasn't so much angry as incredibly surprised. And as I watched him in my rear view mirror slowly slip away, I turned to my instructor and I felt I had to say... "Do you think we should reschedule the test? 'Cause I'm starting now to think it might be best. Either way, I'm pretty sure You could have taught me clutch control In a playground that was emptier than this."
"Jesus Take the Wheel" is metaphorical, but that doesn't keep satirists from pointing out that Jesus' blood has a high alcohol content and that he shouldn't know how to drive.
Vanessa Carlton has a rather grim song called "The Wreckage", where she dreams of either being the cause of a gruesome traffic accident, or one of the victims. Either way, this trope seems to be the cause.
Dr. Bombay's Calcutta (Taxi Taxi Taxi) is about a song about a taxi driver in Calcutta (≧︿≦). Granted, the inference in the song to this fact is him saying he's almost blind and has no license (but he always finds the clutch). Played straighter in the video, however, which shows him, among other things, driving from ''outside'' his driver side window.
Rolling this trope, The Alleged Expert and Car Meets House into a nice single Failure Montage, the video for Rammstein's Benzin is this in spades. With five members of the band taking on the role of a dishevelled-looking firefighter team, the video starts out innocuously enough. Alarm call goes off, they suit up and head down to their fire truck. Then you actually see the damn thing; a colossal ten-wheel drive segmented monster in a parking bay the size of a small hangar. With the whole team sitting five-abreast in the cab, the act of merely driving into the city centre to rescue the rooftop jumper (portrayed by Flake, the band's keyboard player) causes untold chaos. The wheels knock down enough roadside trees to supply a lumber yard, they plough straight through a speeding freight train (the vehicle completely no-selling it, of course), knock a chunk out of a skyscraper by turning a corner, then flip the whole rig as they reach their destination. After showing their solution to the problem - a blanket stretched out between them - the video then cuts to black. Right as the blanket starts to noisily rip right down the middle.
Spike Jones' "Little Bo Peep Has Lost Her Jeep" is an example of either this or Women Drivers. The title character seems not as much an incompetent as she is a total speed freak.