You can catch ants with candy, single piece is totally sufficient, and wont hurt town rating.
You can catch flies with garbage and rotten food. Some say a rotten perfect fruit will do it but I have way better luck with spoiled turnips. Buy just one, let it spoil, then drop it and walk away from it. Once you catch the fly you can then throw it away.
The mole cricket has to be dug up from the ground and then caught with a net. mole cricket is still available but hurry because it disappears in june.
While walking around, you will hear a weird reeeeeeeeeg noise. Start digging around where the sound grows loudest. It can be difficult finding the exact spot.
dream my town, i leave treats and fun stuff