So, I was breeding around ten eggs from my 6IV HA Hoppip in a Dream Ball before I began replacing its egg moves, and I got a freaking sweet shiny Hoppip with 31/xx/31/31/31/31 IVs and +Spd -Atk nature! It doesn't have its HA (got Clorophyl instead, which is really useful anyway), but I guess I could always try and build a team for it.
It's my first ever hatched shiny so I'm really happy that it's one of the pokemon I like, but I wish I could clone it myself so I could have one for doubles and another for singles, though. Getting it without even having the shiny charm or aiming to hatch a shiny makes it even better.