I usually refuse getting signatures from another town as it's too much work. I also refuse giving packages to another villager when that very villager is just walking around so close to the villager who requested it--it's kinda ridiculous, and I wish there was a dialog option for us to say. "He/She's right there beside you, do it yourself you lazy bum!"
I used to give them everything until I joined the forum and realized I was trashing their homes. Lol. Now, I give them furniture to try and restore their homes with original furniture.
I don't give them bugs or fish. Got the petition once but didn't bother with it. I also never make visits even though I try.. I pretty much forget about it or end up doing something else in irl. xD
I usually don't try and catch the exotic fish or bugs if it involves going to the island. I also refuse to catch butterflies in the rain. Come on.. why ask me for that if it's raining? Normally when I go to the island, I do stock up on big beetles, sharks and butterflies so I always have something on hand. Oh and I absolutely REFUSE to get mole crickets. I hate them, hate digging for them, hate everything about them.
Signature ones generally, unless I have time and commitment for them.
I usually accept all requests and not do the ones I don't want to; animals forget about them in 1 day's time anyway. I accept and refuse to complete bug/ fish requests because I know that the animal will be dumb and replace their snowglobe or their bed with it. I'll pretty much do any other request, although I did refuse to buy an Orange for 3000 off Marshal today.
I actually refuse a lot of villager requests except for hide-and-seek games, gathering signatures, having villagers coming over and vise versa, and returning lost items
The scorpion and tarantula are ones I refuse now. Early in the game I gave them both bugs but stopped when a villager wanted it replaced the next day. Those bugs take time to catch.
All the other tasks I do. I have an extra town and usually have to drop something off before I finish playing for the night.
For Catchphrases I say yes and don't change it. The villager thinks it's a new one.
If I fail at a task I always write a letter promising that I'll give them one when I catch it. Write sorry in the letter.
Only gifts in my main town and maybe my second town, I don't want them changing the clothes I want them to wear ;--; and then half of the time they're ugly clothes.
I refuse anything that would mess up their houses - including trading, unless its just wallpaper/ore. i like to give them specific furniture though. I don't keep their houses original, but i do like to decorate them and make them look nice. For example, kiki is my witch's familiar, so i gave her a little cauldron and some clovers, which looks like her little spell book x