I remember having to catch 3 Goletts in Black cos either the ability or nature was bad.
First Golett = Klutz (bad)
Second Golett = Iron Fist (yay!), Modest nature (no!)
Third Golett = Iron Fist, Lax nature (OK)
I evolved the latter into Golurk.
Also, every Pawniard I caught to train had Inner Focus until I caught a fourth one with Defiant and Adamant nature (I had to Synchronize the nature with one of my Elgyems) and I evolved the Defiant Adamant natured female into Bisharp.
Also, the first of those Inner Focus Pawniards had a (dammit!) Calm nature. The second one was Gentle but I knew Adamant was a better nature.
This is why I recommend going on an Elgyem-catching spree so you get different natured Elgyems and hope you also get one with Synchronize.
I remember training up a Pawniard and it landed two Critical Hits in a row on an Audino. And this was only when I was spamming Metal Claw.
Also, I remember catching a Timburr to use against Lenora. It was the first wild Timburr I saw and it was... FEMALE. She since evolved into Gurdurr.
Another moment from Black was that I beat Elesa using a female Zebstrika and I didn't take any major damage.
Now for Shinies, I remember the time in Ruby I caught my first ever Shiny - a Duskull.
In my brother's Diamond he had a Shiny Budew and Unown (U Form).
In SoulSilver I also have two Shinies - a Farfetch'd and Graveler.