Bonus cars
Successfully complete the indicated task to unlock the corresponding car:

Carbon Ikusa GT: Beat all Time Road Rules.
Carson Carbon GT Concept: Find all 400 Smashes.
Carson Fastback: Get a B license.
Hunter Mesquite: Get a D license.
Jansen Carbon X12: Land all 50 Superjumps.
Krieger Carbon Krieger Uberschall 8: Successfully complete two sets of Online Challenges.
Montgomery Carbon Hawker: Hit all 120 Billboards.
Nakamura SI-7: Get a C license.
Rossolini Tempesta: Get an A license.
Watson 25 V16 Revenge Racer: Successfully complete all Road Rage Events.
 

Bonus bikes (Bikes Pack DLC)

Successfully complete the indicated task to unlock the corresponding bike:

Nakumura FV00-TI: Successfully complete 19 Burning or Midnight Rides.
Nakumura Firehawk GP Competition: Successfully complete all Burning and Midnight Rides.
Alternate paint
Successfully complete the indicated task to unlock the corresponding paint:

Gold: Win all Elite License Events.
Platinum: Win all Elite License Events with a 100% completion.
Double barrel roll
A good way to do the double barrel roll is to use the Ikusa GT and fly off the big jump on South Mountain Drive.

Sirens
In any car with sirens (for example, the Hunter Citizen, Krieger PCPD Special, or Hunter Manhattan Spirit), press L3 to turn it on. If you hold it, you will hear a different siren.

Secret quarry area
From the top part of the dam, go west, and follow the road (W. Lake Drive) northwest, and turn left. You should see an opening with a road that leads down and is a very good jump for barrel rolls. The quarry spans that entire area.

Screensaver
Stop your car, and remain idle until the engine turns off and your taillights turn dark. Eventually the screen will become monochrome, and the camera will start to move around the city.

Legendary car references
The following cars are a reference to more famous vehicles:

Carson GT Nighthawk: K.I.T.T. from Knight Rider.
Hunter Cavalry Special: General Lee from The Dukes Of Hazzard.
Hunter Manhattan Special: Ecto 1 from Ghostbusters.
Jansen 88 P12 Special: DeLorean from Back To The Future.

Trophies
Successfully complete one of the following tasks to get a trophy:

Repair your car (Bronze): Awarded for driving through any Auto Repair.
Set a Time Road Rule on any road (Bronze): Awarded for beating any of the original Time Road Rule scores.
Set a Showtime Road Rule on any road (Bronze): Awarded for beating any of the original Showtime Road Rule scores.
Win a Race (Bronze): Awarded for winning any Offline Race.
Smash 3 Burnout Billboards (Bronze): Awarded when you smash through your 3rd pristine Burnout Billboard.
Smash 30 Burnout Billboards (Bronze): Awarded when you smash through your 30th pristine Burnout Billboard.
Drive through 10 sets of Yellow Smash Gates (Bronze): Awarded when you drive through your 10th set of Yellow Smash Gates.
Drive through 50 sets of Yellow Smash Gates (Bronze): Awarded when you drive through your 50th set of Yellow Smash Gates.
Shutdown the Nakamura SI-7 (Bronze): Awarded when you Shutdown the Nakamura SI-7.
Score a x3 Multiplier in Showtime (Bronze): Awarded for reaching a x3 Multiplier in Showtime.
Perform 3 Takedowns (Bronze): Awarded for performing your third Takedown.
Drive through a Repair Shop during a Road Rage (Bronze): Awarded for driving through an Auto Repair during any Offline Road Rage.
Perform a 180 degree Flatspin (Bronze): Awarded for performing a Flatspin of at least 180 degrees.
Successfully land your first Super Jump (Bronze): Awarded when you successfully land your 1st Super Jump.
Score a Boost Chain of x2 (Bronze): Awarded for achieving your first Boost Chain of x2.
Survive a Marked Man (Bronze): Awarded for winning any Offline Marked Man.
Beat the target score in Stunt Run (Bronze): Awarded for winning any Offline Stunt Run.
Power Park your car (Bronze): Awarded for successfully Power Parking your car between two other vehicles.
Set 10 Showtime Road Rules (Bronze): Awarded when you set your 10th Showtime Road Rule.
Set 10 Time Road Rules (Bronze): Awarded when you set your 10th Time Road Rule.
Complete 5 Burning Routes (Bronze): Awarded when you complete your 5th Burning Route.
Drive on the wrong side of the road for 547 yards (Bronze): Awarded for driving a distance of 547 yards in Oncoming Traffic in one go.
Shutdown the Carson Inferno Van (Bronze): Awarded when you Shutdown the Carson Inferno Van.
Successfully land 10 Super Jumps (Bronze): Awarded when you successfully land your 10th Super Jump.
Land a Barrel Roll jump (Bronze): Awarded for successfully landing a Barrel Roll jump.
Find all the Events around Paradise City (Bronze): Awarded when you have discovered every event in Paradise City.
Visit all the Car Parks in Paradise City (Bronze): Awarded when you discover every Car Park in Paradise City.
Invite a friend to join you in Paradise City (Bronze): Awarded for sending an online game invite to another player.
Complete a Freeburn Challenge (Bronze): Awarded when you complete your 1st Freeburn Challenge.
Visit a Gas Station during a Race (Bronze): Awarded for driving through a Gas Station during any Offline Race.
Visit a Paint shop to change your vehicle colour (Bronze): Awarded for driving through any Paint Shop to change your vehicle colour.
Enter a Road Rage in the Carson Inferno Van (Bronze): Awarded for entering any Offline Road Rage in the Carson Inferno Van.
Visit the Airfield (Bronze): Awarded for visiting the Airfield in Silver Lake.
Visit the Quarry (Bronze): Awarded for visiting the Quarry in White Mountain.
Jump Dead Man"s Edge (Bronze): Awarded when you first jump over Dead Man"s Edge in White Mountain.
Jump over another Player"s Car (Bronze): Awarded for jumping over another player"s car in an online game.
8 Players meet in the Wildcats" Baseball Stadium (Bronze): Awarded for meeting up with 7 other players inside the Wildcats" Baseball Stadium in an online game.
Send a Mugshot or Smugshot (Bronze): Awarded for sending a Mugshot or Smughot to another player during an online game (requires a USB camera to be connected).
Take part in an Online Stunt Run (Bronze): Awarded for taking part in an Online Stunt Run.
Take part in an Online Marked Man (Bronze): Awarded for taking part in an Online Marked Man.
Be on the Blue team for Online Road Rage (Bronze): Awarded for taking part in an Online Road Rage as a Runner on the Blue Team.
Be on the Red team for Online Road Rage (Bronze): Awarded for taking part in an Online Road Rage as a Chaser on the Red Team.
Ride 50 Miles Offline on a bike (Bronze): Awarded when you reach a total bike distance ridden of 50 miles.
Complete any Bike Day Road Rule (Bronze): Awarded when you set your 1st bike Day Road Rule.
Complete any Bike Night Road Rule (Bronze): Awarded when you set your 1st bike Night Road Rule.
Ride 10 miles on a bike in an 8 player lobby (Bronze): Awarded for riding 10 miles on a bike in an 8 player online game.
Complete 5 Bike Challenges (Bronze): Awarded when you complete your 5th Freeburn Challenge on a bike.
Drive through 200 sets of Yellow Smash Gates (Silver): Awarded when you drive through your 200th set of Yellow Smash Gates.
Earn your "D" Class License (Silver): Awarded when you receive your "D" Class License.
Earn your "C" Class License (Silver): Awarded when you receive your "C" Class License.
Earn your "B" Class License (Silver): Awarded when you receive your "B" Class License.
Earn your "A" Class License (Silver): Awarded when you receive your "A" Class License.
Score a x5 Multiplier in Showtime (Silver): Awarded for reaching a x5 Multiplier in Showtime.
Score a Boost Chain of x4 (Silver): Awarded for achieving your first Boost Chain of x4.
Score over 50,000 in a Stunt Run (Silver): Awarded for scoring over 50,000 in any Offline Stunt Run.
Drive 750 miles (Silver): Awarded when you reach a total car distance driven of 750 miles.
Complete 10 Freeburn Challenges (Silver): Awarded when you complete your 10th Freeburn Challenge.
Complete your first Timed Challenge (Silver): Awarded when you complete your 1st Timed Challenge.
24 hours in Paradise on a bike (Silver): Awarded for spending a full 24 hour day and night cycle riding a bike.
Smash 60 Burnout Billboards (Gold): Awarded when you smash through your 60th pristine Burnout Billboard.
Earn your "Burnout" Driving License (Gold): Awarded when you receive your "Burnout" Driving License.
Complete 10 Timed Challenges (Gold): Awarded when you complete your 10th Timed Challenge.
Earn 100% on your Bike License (Gold): Awarded when you reach 100% complete on your Bike License.
Burnout Paradise Elite (Platinum): Awarded for successfully collecting all trophies from Burnout Paradise (excluding Additional Content trophies).

The following trophies require the "Party Pack" bonus downloadable content:

Every player take a photo for a Party (Bronze): Awarded when every player in a Party takes a photo of themselves. Requires a USB camera to be connected.
Complete your 1st Party Round (Bronze): Awarded when you complete the 1st Round of your 1st Party.
Every player succeeds in a Party Round (Bronze): Awarded when every player successfully completes a Party Round.
Complete a Massive Party (Silver): Awarded when 8 players have completed an 8 Round Party.
Complete a Big Party (Silver): Awarded when 5-7 players have completed a Party.
Complete a Small Party (Silver): Awarded when 2-4 players have completed a Party.
Every player succeeds in every Party Game (Silver): Awarded upon completing a Party in which every player successfully completes every Game of every Round.
The following trophies require the "Legendary Cars" bonus downloadable content:

Win a Stunt Run in the P12 88 Special (Bronze): Awarded for winning any Offline Stunt Run driving the P12 88 Special.
Light up the Manhattan Spirit during an Online Challenge (Bronze): Awarded for using the flashing lights of the Manhattan Spirit during an Online Freeburn Challenge or Timed Challenge.
Score a Boost Chain of x3 in the GT Nighthawk (Bronze): Awarded for achieving a Boost Chain of x3 driving the GT Nighhawk.
Sound the horn on a Silver Lake Super Jump in the Cavalry Bootlegger (Bronze): Awarded for sounding the horn and successfully landing a Super Jump in Silver Lake driving the Cavalry Bootlegger.
 

The following trophies require the "Toy Cars" bonus downloadable content:

Barrel Roll a Toy Car in the Airfield (Bronze): Awarded for successfully landing a Barrel Roll in the Airfield driving any of the Toy Cars.
Win a Race in a Toy Car (Silver): Awarded for winning any Offline Race driving any of the Toy Cars.
Complete a Bike Timed Challenge using the Toy Bike (Silver): Awarded when you complete any Timed Challenge using the Toy Bike.
The following trophies require the "Boost Specials" bonus downloadable content:

Survive a Marked Man in the Extreme Hot Rod (Silver): Awarded for winning any Offline Marked Man driving the Extreme Hot Rod.
Complete a Timed Challenge using all 3 Boost types in the Hawker Mech (Silver): Awarded when you complete any Timed Challenge driving the Hawker Mech having used all 3 boost types.
The following trophies require the "Cops And Robbers" bonus downloadable content:

Bank the Gold (Silver): Awarded for successfully delivering the Gold to your team base in an Online Cops and Robbers game.
Be the first to reach the Gold (Bronze): Awarded for being the first player in an Online Cops and Robbers game to pick up the Gold.
Pick up dropped Gold (Bronze): Awarded for successfully picking up the Gold once it has been dropped in an Online Cops and Robbers game.
Take down the Gold carrier (Silver): Awarded for successfully performing a Takedown on the player carrying the Gold in an Online Cops and Robbers game.
Win a best of five round game (Silver): Awarded for being on the winning team of a best of five round game of Online Cops and Robbers.
Win a game driving the Watson R-turbo Roadster (Bronze): Awarded for being on the winning team of an Online Cops and Robbers game driving the Watson R-turbo Roadster.
Win a game on the Cops team (Silver): Awarded for being on the winning Cops team in an Online Cops and Robbers game.
Win a game on the Robbers team (Silver): Awarded for being on the winning Robbers team in an Online Cops and Robbers game.