Catch butterflies with a Net:
Caught dead Butterflies can be turned into paintings (frame types: Wooden, Lapis, Iron):
Keep a Jar in your inventory to catch butterflies alive. Gold and Iron Jars break the fastest, Diamond Jars last the longest.
Catch a Cabbage White, a Common Brimstone, a Comma, a Green Hairstreak, an Atlas Moth and a Peacock together with a feather, and ink sac and a book to make a butterfly encyclopedia where you can collect data on the butterflies you caught.
When you right click the Encyclopedia it opens up an Index of all the butterflies you’ve caught so far. Click on a Butterfly for details. Please note that butterflies get only added to your encyclopedia when you’ve crafted the book yourself (not from creative inventory, only then it will display your Name as author in the Tooltip) and only by naturally spawned butterflies (not from the Jar)
For the researching you need first of all a magnifying glass
Then you are ready to do some serious studies and bring it on paper! (Use any butterfly)
Now right click on your findings paper and you might get something like this.
Butterfly research will give you 1, 2 or 3 Fragments of Butterfly spawning data, but only from not yet discovered butterflies of the same family and same or one higher rarity of the butterfly you have studied on your crafting bench (in the upper screenshot, a yellow pansy was studied)
Hint: You can also make a present to your friends and give them high quality findings papers that you find. Getting 3 Fragments at once is quite rare.