It is mostly true that rock and roll musicians no longer compose odes to their beloved automobiles, nor use their preferred make and model of car as a major fixture in their songs. The disdain for the automobile that salimfadhley observes is certainly present in rock music - I'd imagine that it's a stronger presence in Britain, as the best modern anti-car song I can think of is "Killer Cars" by Radiohead. Another recent rock song, "Survival Car" by Fountains Of Wayne, seems to glorify the joy of joyriding on its happy-toned surface, but upon closer examination, its message is more like "driving in the city is a fight for survival". And "Bitchin' Camaro" by the Dead Milkmen likewise portrays a car (a particular type in this case) as an instrument of reckless endangerment.

However, the tradition of namechecking and glorifying one's car is still alive and well in rap, hip-hop, and r&b music. There is a lack of songs about nothing more than the writer's car and why it rules, but the mentioning or use of a car is an essential part of the character of modern rap music.

Some examples:

"In the Ferrari an' Jaguar, switchin' four lanes
Top down, screamin out, 'Money ain't a thang!'"
"Bubble hard in the double R, flash the rings
With the window cracked, holler back, 'Money ain't a thang!'"

-Jermaine Dupri & Jay-Z, respectively, from Money Ain't A Thang

Here, the car is nothing more than a status symbol. The entire song is little more than a well-composed serenade to the institution and privilege of excess, and the cars are an excellent way to press the point.

"Beep beep! Who got the keys to the Jeep?
Vroooooom! I'm drivin' to the beach
Top down, loud sounds, see my peeps"
-Missy Elliott, The Rain (Supa Dupa Fly)

Here, the car is being used to emphasize the playful tone of the lyrics - Jeeps are fun, going to the beach is fun, and making car noises is fun, ergo Missy is fun.

Note that, like in Money Ain't A Thang, she says "top down" - this phrase is a common cliche of 90's rap. Though convertibles are probably far more common today than in years past, they still have the aspect of a status symbol, as they had in the golden years of songs about cars.

Another shared theme in these first two quotes is making noise - be it yelling or just blasting your music ("And then I heard boom from the amplifiers" - from Still Fly by the Big Tymers, which I will come back to later).
This is all part of the idea that the trappings of wealth is the way to privilege - that if you've got a great car and look like you're loaded, you cut loose, fill the neighborhood with sound, do whatever you want to have fun.

"If you wanna go and get high wit' me
Smoke an L in the back of the Benzy-ey"
-Nelly, Ride Wit Me

Here, the car's identity again does not matter as much, since it's just a symbol; but here it's less a symbol of wealth, and almost totally one of privilege, and the ability that wealth or the suggestion of wealth confers on a person. In your luxury car behind your tinted windows, you can drive around all night smoking up with impunity.
Displaying wealth for the sake of image and actually getting pleasure from possessions are two sides of the same coin, though, as both spring from the common rap theme of having something that's better than what you used to.

The Big Tymers, in the two tracks of theirs that I'm familiar with (unfortunately, I only know their hits), use a daunting amount of automobile references. Whether bragging about their own luxuries, or skewering young would-be gangstas who can't afford what they flaunt, the Big Tymers just can't stop talking about cars.

From Still Fly:

"Got a quarter tank of gas
In my new E-Class (In my E-Class Benz)
But that's alright
Cause I'm gon' ride"

And Still Fly goes on to mention Benzes three more times, the Escalade and three other Caddy references, two models of Lex, and the Plymouth Prowler.

This could be dismissed as a mockery of the "hood rich" tendency to mention one's car at any opportunity, but in their earlier track "Get Your Roll On" (which does not seem to be satirical), they refer to "Benzes, Vettes, Hummers, jets" and mention buying five Bentleys as soon as they finish touring... and then they give us this gem of conspicuous bling-blinging:

"Leavin stickers on the Bentley to show the price
Arm out the window just to floss my ice"

...So, taking all that into account, the Big Tymers are just obsessed with their rides, like many of us (especially rappers) are. There's nothing wrong with that, of course, but the fact remains that among some modern musicians, the habit of singing about one's cars is nearly ubiquitous.

Any particularly meaningful, relevant or important car references in modern music? Message me or write them up, please. I've got a little overview of the presence and use of such just in rap here, but it's by no means comprehensive, and I just know I'm missing out on some angles.