Shinies.

[Brendino]

To this day I've managed to catch 3/6 of the shinies that I've come across, not including those like the Red Gyarados.

Most of my shiny luck came on Sapphire, where I caught a shiny Shroomish in the Petalburg Woods on my first playthrough, and then a shiny Heracross in the Safari Zone on the second one. The other shiny that I caught was a Ducklett on the Driftveil Drawbridge on White.

As for improving your luck, there are things like chaining in D/P/Pt and the Shiny Charm in B2/W2 that do improve your odds, but there's still no guarantee that you'll find anything.

 

[Amore]

I've encountered two shinies that weren't a Red Gyarados - a Machoke, when I had no Poké Balls, in Diamond, and a Kricketot on the route between Sandgem Town and Jubilife City on Platinum. Got a few event ones too (3 copies of the shiny Suicune, Entei and Raikou, and a couple of Shiny Pichu, although I've now only got a Gamestop one I traded for).

 

[Aeroblast]

Countless, if I count the ones I RNG'd over the years... maybe 40+?

 

[Sheep]

I haven't caught many.. ;o; My first shiny was Whismur in Sapphire. I also caught a Wurmple that evolved into a Dustox (ugh wanted Beautifly), Volbeat, Rattata, and some others. Got a lot of shinies from a PC member a year ago as well. :)

 

[Brizz]

I've caught a small handful of shinies throughout my playing of Pokemon, including 2 over the past couple of months.

Ponyta (LG)
Unown (Diamond)
Eevee (hatched Diamond)
Bellsprout (LG)
Zigzagoon (Sapphire)
Lunatone (White 2)
Geodude (FR)

 

[d]

Like four, besides the obvious Gyarados. I never actually bothered to try, so the ones I caught were from luck only.

Ponyta, Butterfree, Mareep, Patrat.