Every catchable Pokémon in PTD 1 has a 1/8000 or 0.000125% chance of being found Shiny in the wild. Most Pokémon have had their Shiny forms made available by other means. All Shiny Pokémon in PTD 1 are available via wild, Mystery Gift, SnD Coins, in Game Corner, as prizes for completing Levels, Challenge Levels, or Achievements. You can use SnD Coins to adopt Shiny Pokémon from the Pokémon Adoption, as said atop. They have a different colored scheme compared to Normal Pokémon, and have a sparkling effect. To find a particular Shiny Sprite on this page, use the find function (ctrl+f), to find either by its name or number. (ie. Bulbasaur or 001).
This list shows all of the Pokemon you can encounter in your playthrough of PTD1.