Since March 2016 I've fish chained 86 shinies, I'm going for a living dex and so far I have all but Cloyster and Huntail.
My shortest fish chain is a tie of 4 reels for a Staryu and an Alomomola. The Staryu was actually the first Staryu to appear on the chain.
My longest fish chain before a shiny was 241 for a shiny Basculin. I was actually going for the Dratini, and actually got the 5% shiny Dratini the next time I chained at 61. Only had 8 shiny Basculin and Poliwhirl before Dratini.
My first time fish chaining in Omega Ruby I first got a Magikarp, then a Tentacool, while fishing for a second Tentacool to evolve I caught 6 more shiny Magikarp before Tentacool appeared.
My first ever fish chain was for Corsola, I didn't know about being near rocks so I fished in the middle of the lake, I did have suction cups however. Corsola was my 3rd shiny ever, after 2 random encounter shinies, and it appeared at 97 encounters.
The shiny I have the most of from fish chaining is Skrelp, I have 11. And fished chained 1 Dragalge.