Best: the second week of the game out for a pokemon at midnight on a Friday night, I hadn't found anything interesting, but it was when the game was still really exciting, just got home for the night and a Dragonite spawned right in front of me. I had no idea what I even had but it felt important, I caught it after about 6-8 poke balls / berries (I was only like level 6-8). 336 CP, found out later it had 89 IV SW/DC. Have since powered it up to 2700+ CP and it's still my only dragonite.
Catching a 2800+CP (82IV ZH /HB) Snorlax just outside work during the bonus event deserves an honorable mention. Took 20-30 Ultra Balls + Berries, I thought for sure it was going to run away.
Worst: Bite Twister on my first Gryados (98 IV) after waiting months to finally get one. Got bite / dragon pulse on the second. Also Bite / Bulldoze on my first (98IV) Arcanine. Got Fire Fang /Fire Blast on the second. So I do feel somewhat lucky in terms of not having really bad "worst" moments.