Rare Replay

There are a total of 12 vehicles in Blast Corps and to unlock this achievement you'll need to use all of them at least once. You encounter most of these naturally in the course of a full playthrough of the game, but for reference here's a full list, together with an example location in which the vehicle can be found. Take note in particular of the A-Team Van, which I recommend that you make sure to find as soon as possible, before clearing all stages with gold medals, as once the platinum time trials take over the limited time makes it much harder to reach.

1. Ramdozer: A bulldozer, causing damage by ramming obstacles. Very effective in pushing TNT blocks.
   Available very commonly throughout the game, you'll first find this vehicle in Simian Acres.
2. Backlash: A pick-up truck, causing damage primarily through sliding its armoured back end into obstacles. You'll soon find that this is easier said than done, as the Backlash can be very hard to control effectively.
   The Backlash has its own stage and is first found in the Backlash training stage just after Simian Acres.
3. Skyfall: A rocket car, designed to boost into jumps and then fall onto obstacles to destroy them.
   The Skyfall has its own stage and is first found in the Skyfall training stage adjacent to Havoc District.
4. Thunderfist: A giant robot with a powerful rolling attack - but damaging even by walking into obstacles.
   The Thunderfist has its own stage and is first found in the Thunderfist training stage adjacent to Blackridge Works.
5. Cyclone Suit: Similar to the Thunderfist, but much smaller and with a flip attack rather than a roll.
   The Cyclone Suit doesn't feature very commonly, but can for example be found early in the game, inBlackridge Works.
6. J-Bomb: A jetpack-equipped robot, designed to boost up above obstacles and crash down onto them.
   The J-Bomb has its own stage, first found in the J-Bomb training stage just after Simian Acres.
7. Ballista: A rocket-launching motorcycle; its ranged attack is unusual compared to the other Blast Corpsvehicles.
   The early stage Argent Towers is one of the first places where the Ballista can be found.
8. Sideswipe: An articulated truck, trailing a pair of side-mounted battering rams that spring outwards into obstacles.
   The Sideswipe has its own stage and is first found in the Sideswipe training stage just after Simian Acres.
9. American Dream: A very fast car, red, white and blue in colour. You'll not be using this or any of the cars for much destruction, though.
   The very first stage, Simian Acres, has this vehicle available.
10. Muscle Car: A red/pink-coloured car, much slower than the American Dream but with better cornering. Again, this is not a vehicle designed to cause damage.
    The Muscle Car can be found after a tunnel in Havoc District, or parked on the train in Echo Marches.
11. Police Car: A third car, somewhere between the American Dream and the Muscle Car in terms of speed. As with the other two cars, this is not a vehicle designed to cause damage.
    Agent Towers features the Police Car, after walking down a ramp on foot, blowing up a wall with a TNT crate and taking the train to another area. The Police Car can then later be found in Beeton Tracks.
12. A-Team Van: Trumping the three cars, the black A-Team van is faster even that the American Dream, though still lacks in damage potential.
    The A-Team Van can be found hidden in Crystal Rift, one of the later stages in the game; unless you want to make this very hard for yourself, make a point of locating this before getting gold medals on all stages (after getting all golds, the platinum time trials limit your available time). The Youtube video below demonstrates the van's location, from about 4:00, pushing a black into a tunnel entrance to allow access then following the tunnel to the end.