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