Here is the full list of all 31 Rush Rally Origins achievements.
Unlock 5 cars
Get a gold medal on a Time Trial stage
Get a gold medal on a Race stage
Login to Rush Rally Online and view the leaderboards
Finish first in a championship rally
Finish any stage without hitting anything
Finish any stage without going off the track
Get a gold medal on all stages of a Time Trial rally
Upgrade a car to A-Class
Upgrade 8 cars to A-Class
Unlock all the upgrades for a single car
Unlock all upgrades for 8 cars
Unlock weather on 6 Time Trial rallies
Unlock weather on 6 Race rallies
Unlock weather on all game modes
Get a medal in 6 championship rallies
Finish first in 6 championship rallies
Get a gold medal in time trial without using corner arrows
Get a medal on all stages of 6 rallies in Time Trial
Get a gold medal on all stages of 6 rallies in Time Trial
Get a gold medal on all stages of a Race rally
Get a medal on all stages of 6 rallies in Race
Get a gold medal on all stages of 6 rallies in Race
Get a gold medal on any stage without hitting anything
Get a gold medal on any stage without going off the track
Drive over 500 km
Drive over 1000 km
Hit over 500 objects
Hit over 1000 objects
Download and race against a competitors ghost from the leaderboards
Download and beat a competitors ghost from the leaderboards