Here is the full list of all 37 Rival Stars Horse Racing: Desktop Edition achievements.
Complete a set of 3 races in Online Racing
Complete a set of 3 races in Quick Race
Win after passing 5 horses in the final stretch
Win 10 races in a photo finish
Breed 100 horses
Reach player prestige 20
Win 4 races when riding against Richard
Win 250 races
Reach player prestige 10
Breed 50 horses
Train horses 250 times
Train horses 500 times
Own 12 horses
Own a level 25 horse
Own a level 50 horse
Own a grade 10 horse
Win 50 races
Spend 10,000m in preferred position
Accumulate 1,000,000 Gold
Sell a horse worth more than 15,000 Gold
Own a grade 5 horse
Spend 5,000m in preferred position
Breed a stallion
Breed 10 horses
Train horses 50 times
Own 4 horses
Own 6 horses
Own a level 10 horse
Win 10 races
Win 3 races in a row
Own 2 horses
Own a grade 2 horse
Breed a mare
Reach player prestige 2
Win a race despite a win pay of 15.00 or worse
Win 5 races in Online Racing
Complete a round of Betting Party