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Of course you can get shinies, they've been around since GSC. You can soft reset in front of legendaries/other one-offs, others you have to just find in the wild. And remember, they're rare, around 1/8000 chance.
Actually you can get a shiny in RYB, it'd look normal on there, but if you look at it on GSC it will be shiny. I know someone who caught a gastly and he got on the GBC trade cables and it was shiny.