The next best thing to catching them is to get yourself Butterfly Hutches and then you can have them flittering around in any Building or even Map that you want. In fact, this might be even better as you can see them more easily around your House, etc. than if they were trapped in a tiny terrarium or similar, like the Fish in their Tanks.
On the other hand, if Flying Fish even get added to the game, I want them flying around my Farm when I build the Flying Fish Hutch. Because ridiculous is cool.