Excluding Shinies found while Shiny-hunting something else and Shinies found post-Shiny Charm:
Golbat (Silver) - Thought my eyes were playing tricks on me, but luckily I caught it
Weedle and Pidgey (also Silver)
Koffing (Crystal)
Tentacool (Sapphire)
Minun (Emerald)
Nidoran-M (FireRed) - Lost the cartridge, unfortunately :(
Raticate (Diamond)
Gastly and Machoke (Pearl)
Graveler (also Pearl) - The first Shiny I failed to catch (due to Explosion)
Porygon (Platinum) - I was SRing for a Modest one and got a Shiny instead
Quagsire (also Platinum) - Found in the Great Marsh. It fled...
Machoke and Marill (Platinum again)
Rattata, Hoothoot, Pidgey, and Magnemite (HeartGold)
Weedle (SoulSilver, IIRC)
Red-Stripe Basculin, Litwick, and Chimecho (Black)
Eevee (X) - This one
really surprised me
Diglett (Alpha Sapphire)
Alolan Sanshrew (Moon) - Coincidentally, I had just given up on trying to hatch one
Elekid (Ultra Sun)
Wishiwashi (Sword)
Just for funsies, here's what my team might look like if I only used my accidental Shinies: