When I get lucky and find a shiny pokemon in the wild, that's often pretty cool, whether the pokemon is actually useful or not though is an entirely different story.
As for going out of my way to breed shiny pokemon... it really depends on the pokemon. Having a shiny pokemon provides a sense of uniqueness to the pokemon that most others just don't have. Shiny pokemon that I breed, I breed them shiny when I know that I'm going to be using the pokemon a lot, or if I have a sense of favoritism towards the pokemon, or I think the shiny version is more attractive then the normal version, or if I'm just really attached to the pokemon.
For example... I always demand that my Gorebyss is shiny. My attachment to Gorebyss is mainly focused around a very powerful Gorebyss that was shiny that I had way back in the third generation of games. I became really attached to that particular Gorebyss, and whenever I play with a Gorebyss now I always try to replicate it exactly as the old Gorebyss I used to have. My old Gorebyss was shiny... therefore I always demand that my current Gorebyss is shiny, because it is to make me feel like I've never stopped using the same Gorebyss.
I've had other shiny pokemon across the games I've played, but most of them have never mattered as much as that one shiny golden Gorebyss. Also my favorite color is yellow/gold so everything about a shiny Gorebyss is just perfect to me, if anything the shiny Gorebyss has always been the main Mascot of my team for a fairly long time, and I couldn't imagine not having one in any of the pokemon games I've played (post generation 3).