Even though ultra-rare shiny Pokemon were only properly introduced in Pokemon Gold and Silver, that doesn't mean that they don't exist in Red, Blue, and Yellow, and one mod will make sure you'll never miss any ever again.
Shiny Pokemon are some of the most highly sought-after finds in the entire RPG series. While the odds of finding these alternate color palette Pokemon vary depending on which game you're playing, they can be as low as 1/8192. You might think that the odds would be zero in Red, Blue, and Yellow – after all, you'll never see any sparkles in those games – but that's not quite the case. When transferred to Pokemon Gold, Silver, and Crystal, many Pokemon caught in the original games can actually show up as shiny due to the way that 'shininess' is determined in the second generation.
In the first and second-generation games, all Pokemon are assigned 'DVs,' or 'determinant values' that make them unique. These influence their stats, like their Attack and Defense, much like 'IVs' or 'individual values' do from the third generation onwards, but they also decide whether or not a Pokemon is shiny in Gold, Silver, and Crystal. Even though DVs aren't directly visible in the games, with some math, you can essentially look at a Pokemon's stats in Red, Blue, and Yellow and work out if you've got a rare one on your hands, although the specific stats required can't be found through wild encounters in caves, tall grass, or while surfing.
Having to do math is just one problem, though – the main one is that because you'll never know what a Pokemon's stats are until you've caught it, you could be encountering shinies without even knowing it. Thankfully though, with this mod from NoraInTheFuture, that's no longer the case. With the patch in place, Pokemon that would be shiny in Gen 2 will have a little sparkle symbol next to their name to indicate as such before you've even caught them.
NoraInTheFuture's mod also has an optional setting to modify the RPGs so that their RNG does allow for shiny encounters in the tall grass and so on. The odds are still a measly 1/8192, but hey, at least it makes it possible. It's also recently been updated so that the mod will show the gender of Pokemon in the first-gen games, too – another thing that's not normally visible, but still essentially present in the data for when transferred to the later games. Happy hunting!