i've run into 6 shinies, and thankfully have caught every single one. i'm absolutely terrified of running into a shiny abra or shiny anything that explodes.
first shiny: zubat in emerald. i nearly shat myself. i had to take a moment to process; i didn't even do anything at first. 'is that really a shiny?' 'people actually run into these?' i was so careful not to accidentally hit run away or KO it, and i was so nervous i wasn't going to catch it, but thankfully i did.
second shiny: machop in pearl. i don't really recall this one too well, but it was pretty awesome. it's kind of silly thinking about it now, but i vaguely remember 'oh, another shiny. cool.' after almost losing control of all my bodily functions from the first one, the magic had gone away almost entirely lol.
third shiny: regirock, also in emerald. this was shortly after the 3rd gen RNG had been cracked, and i knew that the RNG just counted up from when you started the game, so while resetting for regirock, i was SRing and just leaving the game on for random periods of time and just took a break to get some food, take a shower, whatever it was i needed to do and then came back to see what had come up and got so excited at SHINY FUCKING REGIROCK that i wasted my masterball on it just not to take any chances. unlike my shiny machop, i felt the rush of excitement when i ran into shiny zubat.
i've caught a few in x, but i believe the safari has a high chance for shinies just from what i've seen (and personally experienced) and one of them was from chain fishing (although none of them were intentional)
first was a shiny pansear in the friend safari. it was the first person i had added after beating the E4, and the first safari i had gone into, and there it was. it was pretty exciting, and i have quite a lot of my catching mons (things with status, false swipe, etc.), so i wasn't really nervous about catching it, but it was a nice way to celebrate finishing the game.
second was a shiny luvdisc i got while trying to farm for heart scales. i only found out later about chainfishing, but even if i had known about it, i got him after a chain of only 5 or so. luvdisc is a pretty adorable shiny... my only problem is no ball really seems to match except the unavailable park ball.
the third and last was a meditite in another safari. i was just looking around to see what was in them, so i wasn't prepared to catch anything. had to throw about 10 dive balls to get him lol. i was just checking real quick before i had to go with my girlfriend and a friend from our building (i don't even recall where we were going), but they were waiting on me, i was just thinking 'just catch it!'
i asked my girlfriend to come over to help, really nervous shy might hit run, so while i was explaining 'just hit this yellow button, and then this one, do NOT UNDER ANY CIRCUMSTANCES HIT THE BLUE BUTTON' i managed to get a critical catch in a dive ball.
'nevermind!' obsessively saved like 5 times, finished getting ready, and we left lol.