I was extremely lucky and caught one with the very first attempt. They only spawn on (in?) clear spots, like dirt or grass so if you have lots of flowers this will make it a bit easier to find the right location to dig.
Go into your inventory and place your shovel first, and your net second. Then use the plus pad to test to see if when you change tools, they're right next to each other in the loop. (Precautionary step.)
Then, go round and round town until you head the low buzzing sort of noise that a mole cricket makes.
When you hear it, walk around in the sound a little until it sounds at its loudest (This means you're really close to it, above ground.)
Once you find where it's generally loudest, start digging around with your shovel until one of the holes produces a large brown bug. At which time the bug buzzing will sound like it's above ground. (This is the mole cricket)
Then quickly switch to your net and go and catch it.
NL has made it (Unintentionally) extremely easy to catch them this time around because of the fast tool switch. Use this to your advantage.
Additionally, after catching it and giving it to the Blathering Owl, go looking for it in the first insect room. You'll hear the buzzing as you enter, and when you walk close to the display above the entrance (along the left side of the display), it pops out and walks around.
It took me a while to catch mine. I happened to visit a town in Spring and it was a big open area. Must've dug up the entire area and it finally showed up. Haven't checked this month.
New Leaf bug bites: 3. Tarantula caught: July 9; Scorpion caught: July 11
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