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