Well, there are no shinies in RBY, so you have a 0% chance of encountering a shiny there.

In GSC, the odds are 1 out of 8194 in random encounters, I believe. But it's easier with breeding if you are using a shiny Pokemon; the odds supposedly increase to 1 in 64! (Source: GameFAQs Guide)

From what I've heard, the odds haven't changed between GSC and RSEFrLg. However, third generation shinies don't have boosted stats like those in GSC, so there's no longer any benefit to catching them for a battling team.