What are the chances of finding a Shiny Magikarp?

The shiny rate appears to have been updated since shiny Magikarp was first released. The observed shiny rate, from The Silph Road is 1 in 450 for random wild encounters. With that data, the observed Magikarp shiny rate was 1 in 360 (95% CI 310 - 440), but testing if there was a difference in rate between all the shiny species collectively was not statistically significant. So it's still assumed to be 1 in 450.

According to The Silph Road, the spawning chance of shiny magikarp is approximately 1/235

Moreover, it seems that the chaining is a thing in pokemon GO too. Chaining is a technique used in the main games where encountering the same pokemon repeatedly increased the chance of a shiny pokemon encounter. Make sure you only enter in combat with magikarp for a while and to capture them.