Here is the full list of all 45 Catch a Falling Star achievements. It takes around 8-10 hours to unlock all of the achievements on Windows.
Score your first 500 points
Score your first 1000 points
Deposit your 100th star
Bump 5 ice with the side of your basket, giving you two StarCoins - don't spend it all in one place ok!
Bump 10 objects with your basket
Buy an increase in space for your basket
Make your first purchase with StarCoins
Purchase a new basket design
Catch/use a "less ice" power-up
Catch/use an "increase basket speed" power-up
Catch a StarCoin
Catch 10 StarCoins
Score your first 100 poitns
Catch/use a "no wind" power-up
Bump ice with the side of your basket
Bump a star with the side of your basket
Bump a star off the side of the screen, saving your life
Deposit your 10th star
Deposit a basket full of stars
Deposit your first star
Deposit a basket that hasn't been completely filled with stars
Start the game and behold how awesome the menu is!
Deposit your 1000th star
Catch a "lose a life" penalty
Catch 10 "lose a life" penalties
Catch/use 10 "increase basket speed" power-ups
Catch/use 50 "increase basket speed" power-ups
Catch/use 100 "increase basket speed" power-ups
Catch 100 StarCoins
Spend 500 StarCoins
Spend 1000 StarCoins
Save 2500 StarCoins
Purchase an in-game power-up with Starcoins
Use an in-game power-up
Buy all the increases in space for your basket
Purchase 10 new basket designs
Purchase all the new basket designs
Unlock the Japan theme
Unlock the Under-the-sea theme
Score your first 2000 points
Score your first 3000 points
Score your first 4000 points
Score your first 4500 points
Score your first 5000 points. NICE!!
Deposit a basket with at least 3 ice